{"id":1588,"date":"2018-01-11T20:28:09","date_gmt":"2018-01-11T20:28:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/exponential-and-logarithmic-functions\/"},"modified":"2018-03-28T16:04:16","modified_gmt":"2018-03-28T16:04:16","slug":"exponential-and-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/chapter\/exponential-and-logarithmic-functions\/","title":{"raw":"1.5 Exponential and Logarithmic Functions","rendered":"1.5 Exponential and Logarithmic Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Identify the form of an exponential function.<\/li>\r\n \t<li>Explain the difference between the graphs of [latex]x^b[\/latex] and [latex]b^x[\/latex].<\/li>\r\n \t<li>Recognize the significance of the number [latex]e[\/latex].<\/li>\r\n \t<li>Identify the form of a logarithmic function.<\/li>\r\n \t<li>Explain the relationship between exponential and logarithmic functions.<\/li>\r\n \t<li>Describe how to calculate a logarithm to a different base.<\/li>\r\n \t<li>Identify the hyperbolic functions, their graphs, and basic identities.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1170572094366\">In this section we examine exponential and logarithmic functions. We use the properties of these functions to solve equations involving exponential or logarithmic terms, and we study the meaning and importance of the number [latex]e[\/latex]. We also define hyperbolic and inverse hyperbolic functions, which involve combinations of exponential and logarithmic functions. (Note that we present alternative definitions of exponential and logarithmic functions in the chapter <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/introduction-6\/\">Applications of Integrations<\/a>, and prove that the functions have the same properties with either definition.)<\/p>\r\n\r\n<div id=\"fs-id1170572108759\" class=\"bc-section section\">\r\n<h1>Exponential Functions<\/h1>\r\n<p id=\"fs-id1170572246259\">Exponential functions arise in many applications. One common example is <span class=\"no-emphasis\">population growth<\/span>.<\/p>\r\n<p id=\"fs-id1170572449480\">For example, if a population starts with [latex]P_0[\/latex] individuals and then grows at an annual rate of [latex]2\\%[\/latex], its population after 1 year is<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]P(1)=P_0+0.02P_0=P_0(1+0.02)=P_0(1.02)[\/latex].<\/div>\r\n<p id=\"fs-id1170572092410\">Its population after 2 years is<\/p>\r\n\r\n<div id=\"fs-id1170572177937\" class=\"equation unnumbered\">[latex]P(2)=P(1)+0.02P(1)=P(1)(1.02)=P_0(1.02)^2[\/latex].<\/div>\r\n<p id=\"fs-id1170572130048\">In general, its population after [latex]t[\/latex] years is<\/p>\r\n\r\n<div id=\"fs-id1170572280288\" class=\"equation unnumbered\">[latex]P(t)=P_0(1.02)^t[\/latex],<\/div>\r\nwhich is an exponential function. More generally, any function of the form [latex]f(x)=b^x[\/latex], where [latex]b&gt;0, \\, b \\ne 1[\/latex], is an exponential function with <strong>base<\/strong> [latex]b[\/latex] and <strong>exponent<\/strong> [latex]x[\/latex]. Exponential functions have constant bases and variable exponents. Note that a function of the form [latex]f(x)=x^b[\/latex] for some constant [latex]b[\/latex] is not an exponential function but a power function.\r\n<p id=\"fs-id1170572248051\">To see the difference between an exponential function and a power function, we compare the functions [latex]y=x^2[\/latex] and [latex]y=2^x[\/latex]. In <a class=\"autogenerated-content\" href=\"#fs-id1170572205233\">(Figure)<\/a>, we see that both [latex]2^x[\/latex] and [latex]x^2[\/latex] approach infinity as [latex]x \\to \\infty[\/latex]. Eventually, however, [latex]2^x[\/latex] becomes larger than [latex]x^2[\/latex] and grows more rapidly as [latex]x \\to \\infty[\/latex]. In the opposite direction, as [latex]x \\to \u2212\\infty, \\, x^2 \\to \\infty[\/latex], whereas [latex]2^x \\to 0[\/latex]. The line [latex]y=0[\/latex] is a horizontal asymptote for [latex]y=2^x[\/latex].<\/p>\r\n\r\n<table id=\"fs-id1170572205233\" class=\"column-header\" style=\"width: 779px\" summary=\"A table with 3 rows and 10 columns. The first row is labeled \u201cx\u201d and has the values \u201c-3; -2; -1; 0; 1; 2; 3; 4; 5; 6\u201d. The second row is labeled \u201cx squared\u201d and has the values \u201c9; 4; 1; 0; 1; 4; 9; 16; 25; 36\u201d. The third row is labeled \u201c2 to the power of x\u201d and has the values \u201c(1\/8); (1\/4); (1\/2); 1; 2; 4; 8; 16; 32; 64\u201d.\"><caption>Values of [latex]x^2[\/latex] and [latex]2^x[\/latex]<\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td style=\"width: 157px\">[latex]\\mathbf{x}[\/latex]<\/td>\r\n<td style=\"width: 138px\">-3<\/td>\r\n<td style=\"width: 138px\">-2<\/td>\r\n<td style=\"width: 138px\">-1<\/td>\r\n<td style=\"width: 9px\">0<\/td>\r\n<td style=\"width: 9px\">1<\/td>\r\n<td style=\"width: 9px\">2<\/td>\r\n<td style=\"width: 9px\">3<\/td>\r\n<td style=\"width: 17px\">4<\/td>\r\n<td style=\"width: 17px\">5<\/td>\r\n<td style=\"width: 17px\">6<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 157px\">[latex]\\mathbf{x^2}[\/latex]<\/td>\r\n<td style=\"width: 138px\">9<\/td>\r\n<td style=\"width: 138px\">4<\/td>\r\n<td style=\"width: 138px\">1<\/td>\r\n<td style=\"width: 9px\">0<\/td>\r\n<td style=\"width: 9px\">1<\/td>\r\n<td style=\"width: 9px\">4<\/td>\r\n<td style=\"width: 9px\">9<\/td>\r\n<td style=\"width: 17px\">16<\/td>\r\n<td style=\"width: 17px\">25<\/td>\r\n<td style=\"width: 17px\">36<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 157px\">[latex]\\mathbf{2^x}[\/latex]<\/td>\r\n<td style=\"width: 138px\">[latex]1\/8[\/latex]<\/td>\r\n<td style=\"width: 138px\">[latex]1\/4[\/latex]<\/td>\r\n<td style=\"width: 138px\">[latex]1\/2[\/latex]<\/td>\r\n<td style=\"width: 9px\">1<\/td>\r\n<td style=\"width: 9px\">2<\/td>\r\n<td style=\"width: 9px\">4<\/td>\r\n<td style=\"width: 9px\">8<\/td>\r\n<td style=\"width: 17px\">16<\/td>\r\n<td style=\"width: 17px\">32<\/td>\r\n<td style=\"width: 17px\">64<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1170572247756\">In <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_001\">(Figure)<\/a>, we graph both [latex]y=x^2[\/latex] and [latex]y=2^x[\/latex] to show how the graphs differ.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_01_05_001\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202644\/CNX_Calc_Figure_01_05_001.jpg\" alt=\"An image of a graph. The x axis runs from -10 to 10 and the y axis runs from 0 to 50. The graph is of two functions. The first function is \u201cy = x squared\u201d, which is a parabola. The function decreases until it hits the origin and then begins increasing. The second function is \u201cy = 2 to the power of x\u201d, which starts slightly above the x axis, and begins increasing very rapidly, more rapidly than the first function.\" width=\"325\" height=\"427\" \/> <strong>Figure 1.<\/strong> Both [latex]2^x[\/latex] and [latex]x^2[\/latex] approach infinity as [latex]x \\to \\infty[\/latex], but [latex]2^x[\/latex] grows more rapidly than [latex]x^2[\/latex]. As [latex]x \\to \u2212\\infty, \\, x^2 \\to \\infty[\/latex], whereas [latex]2^x \\to 0[\/latex].[\/caption]<\/div>\r\n<div id=\"fs-id1170572135350\" class=\"bc-section section\">\r\n<h2>Evaluating Exponential Functions<\/h2>\r\n<p id=\"fs-id1170572134608\">Recall the properties of exponents: If [latex]x[\/latex] is a positive integer, then we define [latex]b^x=b\u00b7b \\cdots b[\/latex] (with [latex]x[\/latex] factors of [latex]b[\/latex]). If [latex]x[\/latex] is a negative integer, then [latex]x=\u2212y[\/latex] for some positive integer [latex]y[\/latex], and we define [latex]b^x=b^{\u2212y}=1\/b^y[\/latex]. Also, [latex]b^0[\/latex] is defined to be 1. If [latex]x[\/latex] is a rational number, then [latex]x=p\/q[\/latex], where [latex]p[\/latex] and [latex]q[\/latex] are integers and [latex]b^x=b^{p\/q}=\\sqrt[q]{b^p}[\/latex]. For example, [latex]9^{3\/2}=\\sqrt{9^3}=27[\/latex]. However, how is [latex]b^x[\/latex] defined if [latex]x[\/latex] is an irrational number? For example, what do we mean by [latex]2^{\\sqrt{2}}[\/latex]? This is too complex a question for us to answer fully right now; however, we can make an approximation. In <a class=\"autogenerated-content\" href=\"#fs-id1170572480690\">(Figure)<\/a>, we list some rational numbers approaching [latex]\\sqrt{2}[\/latex], and the values of [latex]2^x[\/latex] for each rational number [latex]x[\/latex] are presented as well. We claim that if we choose rational numbers [latex]x[\/latex] getting closer and closer to [latex]\\sqrt{2}[\/latex], the values of [latex]2^x[\/latex] get closer and closer to some number [latex]L[\/latex]. We define that number [latex]L[\/latex] to be [latex]2^{\\sqrt{2}}[\/latex].<\/p>\r\n\r\n<table id=\"fs-id1170572480690\" class=\"column-header\" summary=\"A table with 2 rows and 6 columns. The first row is labeled \u201cx\u201d and has the values \u201c1.4; 1.41; 1.414; 1.4142; 1.41421; 1.414213\u201d. The second row is labeled \u201c2 to the power of x\u201d and has the values \u201c2.639; 2.65737; 2.66475; 2.665119; 2.665138; 2.665143\u201d.\"><caption>Values of [latex]2^x[\/latex] for a List of Rational Numbers Approximating [latex]\\sqrt{2}[\/latex]<\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]\\mathbf{x}[\/latex]<\/td>\r\n<td>1.4<\/td>\r\n<td>1.41<\/td>\r\n<td>1.414<\/td>\r\n<td>1.4142<\/td>\r\n<td>1.41421<\/td>\r\n<td>1.414213<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\mathbf{2^x}[\/latex]<\/td>\r\n<td>2.639<\/td>\r\n<td>2.65737<\/td>\r\n<td>2.66475<\/td>\r\n<td>2.665119<\/td>\r\n<td>2.665138<\/td>\r\n<td>2.665143<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"fs-id1170572169644\" class=\"textbox examples\">\r\n<h3>Bacterial Growth<\/h3>\r\n<div id=\"fs-id1170572169649\" class=\"exercise\">\r\n<div id=\"fs-id1170572169651\" class=\"textbox\">\r\n<p id=\"fs-id1170572169653\">Suppose a particular population of bacteria is known to double in size every 4 hours. If a culture starts with 1000 bacteria, the number of bacteria after 4 hours is [latex]n(4)=1000\u00b72[\/latex]. The number of bacteria after 8 hours is [latex]n(8)=n(4)\u00b72=1000\u00b72^2[\/latex]. In general, the number of bacteria after [latex]4m[\/latex] hours is [latex]n(4m)=1000\u00b72^m[\/latex]. Letting [latex]t=4m[\/latex], we see that the number of bacteria after [latex]t[\/latex] hours is [latex]n(t)=1000\u00b72^{t\/4}[\/latex]. Find the number of bacteria after 6 hours, 10 hours, and 24 hours.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572550969\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572550969\"]\r\n<p id=\"fs-id1170572550969\">The number of bacteria after 6 hours is given by [latex]n(6)=1000\u00b72^{6\/4} \\approx 2828[\/latex] bacteria. The number of bacteria after 10 hours is given by [latex]n(10)=1000\u00b72^{10\/4} \\approx 5657[\/latex] bacteria. The number of bacteria after 24 hours is given by [latex]n(24)=1000\u00b72^6=64,000[\/latex] bacteria.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572173708\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572173711\" class=\"exercise\">\r\n<div id=\"fs-id1170572173713\" class=\"textbox\">\r\n<p id=\"fs-id1170572173715\">Given the exponential function [latex]f(x)=100\u00b73^{x\/2}[\/latex], evaluate [latex]f(4)[\/latex] and [latex]f(10)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572173781\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572173781\"]\r\n<p id=\"fs-id1170572173781\">[latex]f(4)=900; \\, f(10)=24,300[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572173826\" class=\"textbox tryit media-2\">\r\n<p id=\"fs-id1170572173829\">Go to <a href=\"http:\/\/www.openstaxcollege.org\/l\/20_exponengrow\">World Population Balance<\/a> for another example of exponential population growth.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572554025\" class=\"bc-section section\">\r\n<h2>Graphing Exponential Functions<\/h2>\r\n<p id=\"fs-id1170572554031\">For any base [latex]b&gt;0, \\, b\\ne 1[\/latex], the exponential function [latex]f(x)=b^x[\/latex] is defined for all real numbers [latex]x[\/latex] and [latex]b^x&gt;0[\/latex]. Therefore, the domain of [latex]f(x)=b^x[\/latex] is [latex](\u2212\\infty ,\\infty)[\/latex] and the range is [latex](0,\\infty)[\/latex]. To graph [latex]b^x[\/latex], we note that for [latex]b&gt;1, \\, b^x[\/latex] is increasing on [latex](\u2212\\infty ,\\infty)[\/latex] and [latex]b^x \\to \\infty [\/latex] as [latex]x \\to \\infty[\/latex], whereas [latex]b^x \\to 0[\/latex] as [latex]x \\to \u2212\\infty[\/latex]. On the other hand, if [latex]0&lt;b&lt;1, \\, f(x)=b^x[\/latex] is decreasing on [latex](\u2212\\infty ,\\infty)[\/latex] and [latex]b^x \\to 0[\/latex] as [latex]x \\to \\infty [\/latex] whereas [latex]b^x \\to \\infty [\/latex] as [latex]x \\to \u2212\\infty [\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_002\">(Figure)<\/a>).<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_01_05_002\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202647\/CNX_Calc_Figure_01_05_002.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 3 and the y axis runs from 0 to 4. The graph is of four functions. The first function is \u201cf(x) = 2 to the power of x\u201d, an increasing curved function, which starts slightly above the x axis and begins increasing. The second function is \u201cf(x) = 4 to the power of x\u201d, an increasing curved function, which starts slightly above the x axis and begins increasing rapidly, more rapidly than the first function. The third function is \u201cf(x) = (1\/2) to the power of x\u201d, a decreasing curved function with decreases until it gets close to the x axis without touching it. The third function is \u201cf(x) = (1\/4) to the power of x\u201d, a decreasing curved function with decreases until it gets close to the x axis without touching it. It decrases at a faster rate than the third function.\" width=\"325\" height=\"221\" \/> <strong>Figure 2.<\/strong> If [latex]b&gt;1[\/latex], then [latex]b^x[\/latex] is increasing on [latex](\u2212\\infty ,\\infty)[\/latex]. If [latex]0&lt;b&lt;1[\/latex], then [latex]b^x[\/latex] is decreasing on [latex](\u2212\\infty ,\\infty)[\/latex].[\/caption]<\/div>\r\n<div id=\"fs-id1170572481215\" class=\"textbox tryit media-2\">\r\n<p id=\"fs-id1170572481218\">Visit this <a href=\"http:\/\/www.openstaxcollege.org\/l\/20_inverse\">site<\/a> for more exploration of the graphs of exponential functions.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1170572481226\">Note that exponential functions satisfy the general laws of exponents. To remind you of these laws, we state them as rules.<\/p>\r\n\r\n<div id=\"fs-id1170572481231\" class=\"textbox key-takeaways\">\r\n<h3>Rule: Laws of Exponents<\/h3>\r\n<p id=\"fs-id1170572481236\">For any constants [latex]a&gt;0, \\, b&gt;0[\/latex], and for all [latex]x[\/latex] and [latex]y[\/latex],<\/p>\r\n\r\n<ol id=\"fs-id1170572481268\">\r\n \t<li>[latex]b^x\u00b7b^y=b^{x+y}[\/latex]<\/li>\r\n \t<li>[latex]\\large\\frac{b^x}{b^y} \\normalsize = b^{x-y}[\/latex]<\/li>\r\n \t<li>[latex](b^x)^y=b^{xy}[\/latex]<\/li>\r\n \t<li>[latex](ab)^x=a^x b^x[\/latex]<\/li>\r\n \t<li>[latex]\\large \\frac{a^x}{b^x} \\normalsize =(\\frac{a}{b})^x[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1170572440093\" class=\"textbox examples\">\r\n<h3>Using the Laws of Exponents<\/h3>\r\n<div id=\"fs-id1170572440098\" class=\"exercise\">\r\n<div id=\"fs-id1170572440100\" class=\"textbox\">\r\n<p id=\"fs-id1170572440102\">Use the laws of exponents to simplify each of the following expressions.<\/p>\r\n\r\n<ol id=\"fs-id1170572440106\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\large \\frac{(2x^{2\/3})^3}{(4x^{-1\/3})^2}[\/latex]<\/li>\r\n \t<li>[latex]\\large \\frac{(x^3 y^{-1})^2}{(xy^2)^{-2}}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572453127\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572453127\"]\r\n<ol id=\"fs-id1170572453127\" style=\"list-style-type: lower-alpha\">\r\n \t<li>We can simplify as follows:\r\n<div id=\"fs-id1170570966957\" class=\"equation unnumbered\">[latex]\\large \\frac{(2x^{2\/3})^3}{(4x^{-1\/3})^2} \\normalsize = \\large \\frac{2^3(x^{2\/3})^3}{4^2(x^{-1\/3})^2} \\normalsize = \\large \\frac{8x^2}{16x^{-2\/3}} \\normalsize = \\large \\frac{x^2x^{2\/3}}{2} \\normalsize = \\large \\frac{x^{8\/3}}{2}[\/latex].<\/div><\/li>\r\n \t<li>We can simplify as follows:\r\n<div id=\"fs-id1170573582280\" class=\"equation unnumbered\">[latex]\\large \\frac{(x^3y^{-1})^2}{(xy^2)^{-2}} \\normalsize = \\large \\frac{(x^3)^2(y^{-1})^2}{x^{-2}(y^2)^{-2}} \\normalsize = \\large \\frac{x^6y^{-2}}{x^{-2}y^{-4}} \\normalsize = x^6x^2y^{-2}y^4 = x^8y^2[\/latex].<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises checkpoint\">\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1170572452234\">Use the laws of exponents to simplify [latex](6x^{-3}y^2)\/(12x^{-4}y^5)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170572452533\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572452533\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572452533\"][latex]x\/(2y^3)[\/latex]<\/div>\r\n<div id=\"fs-id1165042836022\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042707513\">[latex]x^a\/x^b=x^{a-b}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572452563\" class=\"bc-section section\">\r\n<h1>The <strong>Number<\/strong><em><strong>\u00a0[latex]e[\/latex]<\/strong><\/em><\/h1>\r\n<p id=\"fs-id1170572452572\">A special type of exponential function appears frequently in real-world applications. To describe it, consider the following example of exponential growth, which arises from <span class=\"no-emphasis\">compounding interest<\/span> in a savings account. Suppose a person invests [latex]P[\/latex] dollars in a savings account with an annual interest rate [latex]r[\/latex], compounded annually. The amount of money after 1 year is<\/p>\r\n\r\n<div id=\"fs-id1170572452592\" class=\"equation unnumbered\">[latex]A(1)=P+rP=P(1+r)[\/latex].<\/div>\r\n<p id=\"fs-id1170572449156\">The amount of money after 2 years is<\/p>\r\n\r\n<div id=\"fs-id1170572449164\" class=\"equation unnumbered\">[latex]A(2)=A(1)+rA(1)=P(1+r)+rP(1+r)=P(1+r)^2[\/latex].<\/div>\r\n<p id=\"fs-id1170572449268\">More generally, the amount after [latex]t[\/latex] years is<\/p>\r\n\r\n<div id=\"fs-id1170572449276\" class=\"equation unnumbered\">[latex]A(t)=P(1+r)^t[\/latex].<\/div>\r\n<p id=\"fs-id1170572449316\">If the money is compounded 2 times per year, the amount of money after half a year is<\/p>\r\n\r\n<div id=\"fs-id1170572449320\" class=\"equation unnumbered\">[latex]A(\\frac{1}{2})=P+(\\frac{r}{2})P=P(1+(\\frac{r}{2}))[\/latex].<\/div>\r\n<p id=\"fs-id1170572548410\">The amount of money after 1 year is<\/p>\r\n\r\n<div id=\"fs-id1170572548417\" class=\"equation unnumbered\">[latex]A(1)=A(\\frac{1}{2})+(\\frac{r}{2})A(\\frac{1}{2})=P(1+\\frac{r}{2})+\\frac{r}{2}(P(1+\\frac{r}{2}))=P(1+\\frac{r}{2})^2[\/latex].<\/div>\r\n<p id=\"fs-id1170572229281\">After [latex]t[\/latex] years, the amount of money in the account is<\/p>\r\n\r\n<div id=\"fs-id1170572229288\" class=\"equation unnumbered\">[latex]A(t)=P(1+\\frac{r}{2})^{2t}[\/latex].<\/div>\r\n<p id=\"fs-id1170572229335\">More generally, if the money is compounded [latex]n[\/latex] times per year, the amount of money in the account after [latex]t[\/latex] years is given by the function<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]A(t)=P(1+\\frac{r}{n})^{nt}[\/latex].<\/div>\r\n<p id=\"fs-id1170572229394\">What happens as [latex]n\\to \\infty[\/latex]? To answer this question, we let [latex]m=n\/r[\/latex] and write<\/p>\r\n\r\n<div id=\"fs-id1170572451284\" class=\"equation unnumbered\">[latex](1+\\frac{r}{n})^{nt}=(1+\\frac{1}{m})^{mrt}[\/latex],<\/div>\r\n<p id=\"fs-id1170572451344\">and examine the behavior of [latex](1+1\/m)^m[\/latex] as [latex]m\\to \\infty[\/latex], using a table of values (<a class=\"autogenerated-content\" href=\"#fs-id1170572451390\">(Figure)<\/a>).<\/p>\r\n\r\n<table id=\"fs-id1170572451390\" class=\"column-header\" summary=\"A table with 2 rows and 6 columns. The first row is labeled \u201cm\u201d and has the values \u201c10; 100; 1000; 10,000; 100,000; 1,000,000\u201d. The second row is labeled \u201c(1 + (1\/m)) to the power of m\u201d and has the values \u201c2.5937; 2.7048; 2.71692; 2.71815; 2.718268; 2.718280\u201d.\"><caption>Values of [latex](1+\\frac{1}{m})^m[\/latex] as [latex]m \\to \\infty [\/latex]<\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]\\mathbf{m}[\/latex]<\/td>\r\n<td>10<\/td>\r\n<td>100<\/td>\r\n<td>1000<\/td>\r\n<td>10,000<\/td>\r\n<td>100,000<\/td>\r\n<td>1,000,000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\mathbf{(1+\\frac{1}{m})^m}[\/latex]<\/td>\r\n<td>2.5937<\/td>\r\n<td>2.7048<\/td>\r\n<td>2.71692<\/td>\r\n<td>2.71815<\/td>\r\n<td>2.718268<\/td>\r\n<td>2.718280<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLooking at this table, it appears that [latex](1+1\/m)^m[\/latex] is approaching a number between 2.7 and 2.8 as [latex]m\\to \\infty [\/latex]. In fact, [latex](1+1\/m)^m[\/latex] does approach some number as [latex]m\\to \\infty [\/latex]. We call this number [latex]e[\/latex]. To six decimal places of accuracy,\r\n<div id=\"fs-id1170572549050\" class=\"equation unnumbered\">[latex]e \\approx 2.718282[\/latex].<\/div>\r\n<p id=\"fs-id1170572549064\">The letter [latex]e[\/latex] was first used to represent this number by the Swiss mathematician Leonhard Euler during the 1720s. Although Euler did not discover the number, he showed many important connections between [latex]e[\/latex] and logarithmic functions. We still use the notation [latex]e[\/latex] today to honor Euler\u2019s work because it appears in many areas of mathematics and because we can use it in many practical applications.<\/p>\r\n<p id=\"fs-id1170572549084\">Returning to our savings account example, we can conclude that if a person puts [latex]P[\/latex] dollars in an account at an annual interest rate [latex]r[\/latex], compounded continuously, then [latex]A(t)=Pe^{rt}[\/latex]. This function may be familiar. Since functions involving base [latex]e[\/latex] arise often in applications, we call the function [latex]f(x)=e^x[\/latex] the <strong>natural exponential function<\/strong>. Not only is this function interesting because of the definition of the number [latex]e[\/latex], but also, as discussed next, its graph has an important property.<\/p>\r\n<p id=\"fs-id1170572451705\">Since [latex]e&gt;1[\/latex], we know [latex]e^x[\/latex] is increasing on [latex](\u2212\\infty ,\\infty)[\/latex]. In <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_003\">(Figure)<\/a>, we show a graph of [latex]f(x)=e^x[\/latex] along with a <em>tangent line<\/em> to the graph of at [latex]x=0[\/latex]. We give a precise definition of tangent line in the next chapter; but, informally, we say a tangent line to a graph of [latex]f[\/latex] at [latex]x=a[\/latex] is a line that passes through the point [latex](a,f(a))[\/latex] and has the same \u201cslope\u201d as [latex]f[\/latex] at that point. The function [latex]f(x)=e^x[\/latex] is the only exponential function [latex]b^x[\/latex] with tangent line at [latex]x=0[\/latex] that has a slope of 1. As we see later in the text, having this property makes the natural exponential function the simplest exponential function to use in many instances.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_01_05_003\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202649\/CNX_Calc_Figure_01_05_003.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 3 and the y axis runs from 0 to 4. The graph is of the function \u201cf(x) = e to power of x\u201d, an increasing curved function that starts slightly above the x axis. The y intercept is at the point (0, 1). At this point, a line is drawn tangent to the function. This line has the label \u201cslope = 1\u201d.\" width=\"325\" height=\"202\" \/> <strong>Figure 3.<\/strong> The graph of [latex]f(x)=e^x[\/latex] has a tangent line with slope 1 at [latex]x=0[\/latex].[\/caption]<\/div>\r\n<div id=\"fs-id1170572547818\" class=\"textbox examples\">\r\n<h3>Compounding Interest<\/h3>\r\n<div id=\"fs-id1170572547823\" class=\"exercise\">\r\n<div id=\"fs-id1170572547825\" class=\"textbox\">\r\n<p id=\"fs-id1170572547827\">Suppose [latex]\\$500[\/latex] is invested in an account at an annual interest rate of [latex]r=5.5\\%[\/latex], compounded continuously.<\/p>\r\n\r\n<ol id=\"fs-id1170572547852\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Let [latex]t[\/latex] denote the number of years after the initial investment and [latex]A(t)[\/latex] denote the amount of money in the account at time [latex]t[\/latex]. Find a formula for [latex]A(t)[\/latex].<\/li>\r\n \t<li>Find the amount of money in the account after 10 years and after 20 years.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572542850\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572542850\"]\r\n<ol id=\"fs-id1170572542850\" style=\"list-style-type: lower-alpha\">\r\n \t<li>If [latex]P[\/latex] dollars are invested in an account at an annual interest rate [latex]r[\/latex], compounded continuously, then [latex]A(t)=Pe^{rt}[\/latex]. Here [latex]P=\\$500[\/latex] and [latex]r=0.055[\/latex]. Therefore, [latex]A(t)=500e^{0.055t}[\/latex].<\/li>\r\n \t<li>After 10 years, the amount of money in the account is\r\n<div id=\"fs-id1170573386623\" class=\"equation unnumbered\">[latex]A(10)=500e^{0.055\u00b710}=500e^{0.55}\\approx \\$866.63[\/latex].<\/div>\r\nAfter 20 years, the amount of money in the account is\r\n<div id=\"fs-id1170573390418\" class=\"equation unnumbered\">[latex]A(20)=500e^{0.055\u00b720}=500e^{1.1}\\approx \\$1,502.08[\/latex].<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572455119\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572455122\" class=\"exercise\">\r\n<div id=\"fs-id1170572455124\" class=\"textbox\">\r\n<p id=\"fs-id1170572455126\">If [latex]\\$750[\/latex] is invested in an account at an annual interest rate of [latex]4\\%[\/latex], compounded continuously, find a formula for the amount of money in the account after [latex]t[\/latex] years. Find the amount of money after 30 years.<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<div class=\"solution\">\r\n<p id=\"fs-id1170572455190\">[reveal-answer q=\"505690\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"505690\"][latex]A(t)=750e^{0.04t}[\/latex]. After 30 years, there will be approximately [latex]\\$2,490.09[\/latex].[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165039562082\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042707350\">[latex]A(t)=Pe^{rt}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572455243\" class=\"bc-section section\">\r\n<h1>Logarithmic Functions<\/h1>\r\n<p id=\"fs-id1170572455248\">Using our understanding of exponential functions, we can discuss their inverses, which are the logarithmic functions. These come in handy when we need to consider any phenomenon that varies over a wide range of values, such as pH in chemistry or decibels in sound levels.<\/p>\r\n<p id=\"fs-id1170572455254\">The exponential function [latex]f(x)=b^x[\/latex] is one-to-one, with domain [latex](\u2212\\infty ,\\infty)[\/latex] and range [latex](0,\\infty )[\/latex]. Therefore, it has an inverse function, called the <em>logarithmic function with base<\/em> [latex]b[\/latex]. For any [latex]b&gt;0, \\, b \\ne 1[\/latex], the logarithmic function with base [latex]b[\/latex], denoted [latex]\\log_b[\/latex], has domain [latex](0,\\infty )[\/latex] and range [latex](\u2212\\infty ,\\infty )[\/latex], and satisfies<\/p>\r\n\r\n<div id=\"fs-id1170572551548\" class=\"equation unnumbered\">[latex]\\log_b(x)=y[\/latex] if and only if [latex]b^y=x[\/latex].<\/div>\r\n<p id=\"fs-id1170572545100\">For example,<\/p>\r\n\r\n<div id=\"fs-id1170572545103\" class=\"equation unnumbered\">[latex]\\begin{array}{cccc} \\log_2 (8)=3\\hfill &amp; &amp; &amp; \\text{since}\\phantom{\\rule{3em}{0ex}}2^3=8,\\hfill \\\\ \\log_{10} (\\frac{1}{100})=-2\\hfill &amp; &amp; &amp; \\text{since}\\phantom{\\rule{3em}{0ex}}10^{-2}=\\frac{1}{10^2}=\\frac{1}{100},\\hfill \\\\ \\log_b (1)=0\\hfill &amp; &amp; &amp; \\text{since}\\phantom{\\rule{3em}{0ex}}b^0=1 \\, \\text{for any base} \\, b&gt;0.\\hfill \\end{array}[\/latex]<\/div>\r\nFurthermore, since [latex]y=\\log_b (x)[\/latex] and [latex]y=b^x[\/latex] are inverse functions,\r\n<div id=\"fs-id1170572169121\" class=\"equation unnumbered\">[latex]\\log_b (b^x)=x \\, \\text{and} \\, b^{\\log_b (x)}=x[\/latex].<\/div>\r\n<p id=\"fs-id1170572169184\">The most commonly used logarithmic function is the function [latex]\\log_e (x)[\/latex]. Since this function uses natural [latex]e[\/latex] as its base, it is called the<strong> natural logarithm<\/strong>. Here we use the notation [latex]\\ln(x)[\/latex] or [latex]\\ln x[\/latex] to mean [latex]\\log_e (x)[\/latex]. For example,<\/p>\r\n\r\n<div id=\"fs-id1170572243723\" class=\"equation unnumbered\">[latex]\\ln (e)=\\log_e (e)=1, \\, \\ln(e^3)=\\log_e (e^3)=3, \\, \\ln(1)=\\log_e (1)=0[\/latex].<\/div>\r\n<p id=\"fs-id1170572243836\">Since the functions [latex]f(x)=e^x[\/latex] and [latex]g(x)=\\ln(x)[\/latex] are inverses of each other,<\/p>\r\n\r\n<div id=\"fs-id1170572243887\" class=\"equation unnumbered\">[latex]\\ln(e^x)=x \\, \\text{and} \\, e^{\\ln x}=x[\/latex],<\/div>\r\n<p id=\"fs-id1170572229068\">and their graphs are symmetric about the line [latex]y=x[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_004\">(Figure)<\/a>).<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_01_05_004\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202652\/CNX_Calc_Figure_01_05_004.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -3 to 4. The graph is of two functions. The first function is \u201cf(x) = e to power of x\u201d, an increasing curved function that starts slightly above the x axis. The y intercept is at the point (0, 1) and there is no x intercept. The second function is \u201cf(x) = ln(x)\u201d, an increasing curved function. The x intercept is at the point (1, 0) and there is no y intercept. A dotted line with label \u201cy = x\u201d is also plotted on the graph, to show that the functions are mirror images over this line.\" width=\"325\" height=\"312\" \/> Figure 4: The functions [latex]y=e^x[\/latex] and [latex]y=\\ln(x)[\/latex] are inverses of each other, so their graphs are symmetric about the line [latex]y=x[\/latex].[\/caption]<\/div>\r\n<div id=\"fs-id1170572229152\" class=\"textbox tryit media-2\">\r\n<p id=\"fs-id1170572229156\">At this <a href=\"http:\/\/www.openstaxcollege.org\/l\/20_logscale\">site<\/a> you can see an example of a base-10 logarithmic scale.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1170572229166\">In general, for any base [latex]b&gt;0, \\, b\\ne 1[\/latex], the function [latex]g(x)=\\log_b (x)[\/latex] is symmetric about the line [latex]y=x[\/latex] with the function [latex]f(x)=b^x[\/latex]. Using this fact and the graphs of the exponential functions, we graph functions [latex]\\log_b (x)[\/latex] for several values of [latex]b&gt;1[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_005\">(Figure)<\/a>).<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_01_05_005\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202654\/CNX_Calc_Figure_01_05_005.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 3 and the y axis runs from 0 to 4. The graph is of three functions. All three functions a log functions that are increasing curved functions that start slightly to the right of the y axis and have an x intercept at (1, 0). The first function is \u201cy = log base 10 (x)\u201d, the second function is \u201cf(x) = ln(x)\u201d, and the third function is \u201cy = log base 2 (x)\u201d. The third function increases the most rapidly, the second function increases next most rapidly, and the third function increases the slowest.\" width=\"325\" height=\"312\" \/> Figure 5: Graphs of [latex]y=\\log_b (x)[\/latex] are depicted for [latex]b=2, \\, e, \\, 10[\/latex].[\/caption]<\/div>\r\n<p id=\"fs-id1170572482697\">Before solving some equations involving exponential and logarithmic functions, let\u2019s review the basic properties of logarithms.<\/p>\r\n\r\n<div id=\"fs-id1170572482702\" class=\"textbox key-takeaways\">\r\n<h3>Rule: Properties of Logarithms<\/h3>\r\n<p id=\"fs-id1170572482707\">If [latex]a,b,c&gt;0, \\, b\\ne 1[\/latex], and [latex]r[\/latex] is any real number, then<\/p>\r\n\r\n<div id=\"fs-id1170572482744\" class=\"equation unnumbered\">[latex]\\begin{array}{cccc}1.\\phantom{\\rule{2em}{0ex}}\\log_b (ac)=\\log_b (a)+\\log_b (c)\\hfill &amp; &amp; &amp; \\text{(Product property)}\\hfill \\\\ 2.\\phantom{\\rule{2em}{0ex}}\\log_b(\\frac{a}{c})=\\log_b (a) -\\log_b (c)\\hfill &amp; &amp; &amp; \\text{(Quotient property)}\\hfill \\\\ 3.\\phantom{\\rule{2em}{0ex}}\\log_b (a^r)=r \\log_b (a)\\hfill &amp; &amp; &amp; \\text{(Power property)}\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572551970\" class=\"textbox examples\">\r\n<h3>Solving Equations Involving Exponential Functions<\/h3>\r\n<div id=\"fs-id1170572551976\" class=\"exercise\">\r\n<div id=\"fs-id1170572551978\" class=\"textbox\">\r\n<p id=\"fs-id1170572551980\">Solve each of the following equations for [latex]x[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1170572551988\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]5^x=2[\/latex]<\/li>\r\n \t<li>[latex]e^x+6e^{\u2212x}=5[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572550555\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572550555\"]\r\n<ol id=\"fs-id1170572550555\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Applying the natural logarithm function to both sides of the equation, we have\r\n<div id=\"fs-id1170571071223\" class=\"equation unnumbered\">[latex]\\ln 5^x=\\ln 2[\/latex].<\/div>\r\nUsing the power property of logarithms,\r\n<div id=\"fs-id1170571277779\" class=\"equation unnumbered\">[latex]x \\ln 5=\\ln 2[\/latex].<\/div>\r\nTherefore, [latex]x=\\ln 2 \/ \\ln 5[\/latex].<\/li>\r\n \t<li>Multiplying both sides of the equation by [latex]e^x[\/latex], we arrive at the equation\r\n<div id=\"fs-id1170571301573\" class=\"equation unnumbered\">[latex]e^{2x}+6=5e^x[\/latex].<\/div>\r\nRewriting this equation as\r\n<div id=\"fs-id1170573367583\" class=\"equation unnumbered\">[latex]e^{2x}-5e^x+6=0[\/latex],<\/div>\r\nwe can then rewrite it as a quadratic equation in [latex]e^x[\/latex]:\r\n<div id=\"fs-id1170570976384\" class=\"equation unnumbered\">[latex](e^x)^2-5(e^x)+6=0[\/latex].<\/div>\r\nNow we can solve the quadratic equation. Factoring this equation, we obtain\r\n<div id=\"fs-id1170573400246\" class=\"equation unnumbered\">[latex](e^x-3)(e^x-2)=0[\/latex].<\/div>\r\nTherefore, the solutions satisfy [latex]e^x=3[\/latex] and [latex]e^x=2[\/latex]. Taking the natural logarithm of both sides gives us the solutions [latex]x=\\ln 3, \\, \\ln 2[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572232044\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572232048\" class=\"exercise\">\r\n<div id=\"fs-id1170572232051\" class=\"textbox\">\r\n<p id=\"fs-id1170572232053\">Solve [latex]e^{2x} \/ (3+e^{2x})=1\/2[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572174654\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572174654\"]\r\n<p id=\"fs-id1170572174654\">[latex]x=\\frac{\\ln 3}{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042579088\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042579095\">First solve the equation for [latex]e^{2x}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572174674\" class=\"textbox examples\">\r\n<h3>Solving Equations Involving Logarithmic Functions<\/h3>\r\n<div id=\"fs-id1170572174679\" class=\"exercise\">\r\n<div id=\"fs-id1170572174682\" class=\"textbox\">\r\n<p id=\"fs-id1170572174684\">Solve each of the following equations for [latex]x[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1170572174692\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\ln (\\frac{1}{x})=4[\/latex]<\/li>\r\n \t<li>[latex]\\log_{10} \\sqrt{x}+ \\log_{10} x=2[\/latex]<\/li>\r\n \t<li>[latex]\\ln(2x)-3 \\ln(x^2)=0[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572174799\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572174799\"]\r\n<ol id=\"fs-id1170572174799\" style=\"list-style-type: lower-alpha\">\r\n \t<li>By the definition of the natural logarithm function,\r\n<div id=\"fs-id1170573425282\" class=\"equation unnumbered\">[latex]\\ln\\big(\\frac{1}{x}\\big)=4 \\, \\text{if and only if} \\, e^4=\\frac{1}{x}[\/latex].<\/div>\r\nTherefore, the solution is [latex]x=1\/e^4[\/latex].<\/li>\r\n \t<li>Using the product and power properties of logarithmic functions, rewrite the left-hand side of the equation as\r\n<div id=\"fs-id1170573416245\" class=\"equation unnumbered\">[latex]\\log_{10} \\sqrt{x}+ \\log_{10} x = \\log_{10} x \\sqrt{x} = \\log_{10}x^{3\/2} = \\frac{3}{2} \\log_{10} x[\/latex].<\/div>\r\nTherefore, the equation can be rewritten as\r\n<div id=\"fs-id1170571053549\" class=\"equation unnumbered\">[latex]\\frac{3}{2} \\log_{10} x = 2 \\, \\text{or} \\, \\log_{10} x = \\frac{4}{3}[\/latex].<\/div>\r\nThe solution is [latex]x=10^{4\/3}=10\\sqrt[3]{10}[\/latex].<\/li>\r\n \t<li>Using the power property of logarithmic functions, we can rewrite the equation as [latex]\\ln(2x) - \\ln(x^6) = 0[\/latex].\r\nUsing the quotient property, this becomes\r\n<div id=\"fs-id1170573426389\" class=\"equation unnumbered\">[latex]\\ln\\big(\\frac{2}{x^5}\\big)=0[\/latex].<\/div>\r\nTherefore, [latex]2\/x^5=1[\/latex], which implies [latex]x=\\sqrt[5]{2}[\/latex]. We should then check for any extraneous solutions.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572552646\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572552650\" class=\"exercise\">\r\n<div id=\"fs-id1170572552652\" class=\"textbox\">\r\n<p id=\"fs-id1170572552654\">Solve [latex]\\ln(x^3)-4 \\ln (x)=1[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572552698\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572552698\"]\r\n<p id=\"fs-id1170572552698\">[latex]x=\\frac{1}{e}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165043161242\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165043161250\">First use the power property, then use the product property of logarithms.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572552715\">When evaluating a logarithmic function with a calculator, you may have noticed that the only options are [latex]\\log_{10}[\/latex] or log, called the <span class=\"no-emphasis\"><em>common logarithm<\/em><\/span>, or ln, which is the natural logarithm. However, exponential functions and logarithm functions can be expressed in terms of any desired base [latex]b[\/latex]. If you need to use a calculator to evaluate an expression with a different base, you can apply the change-of-base formulas first. Using this change of base, we typically write a given exponential or logarithmic function in terms of the natural exponential and natural logarithmic functions.<\/p>\r\n\r\n<div id=\"fs-id1170572552753\" class=\"textbox key-takeaways\">\r\n<h3>Rule: Change-of-Base Formulas<\/h3>\r\n<p id=\"fs-id1170572267920\">Let [latex]a&gt;0, \\, b&gt;0[\/latex], and [latex]a\\ne 1, \\, b\\ne 1[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1170572267962\">\r\n \t<li>[latex]a^x=b^{x \\log_b a}[\/latex] for any real number [latex]x[\/latex].\r\nIf [latex]b=e[\/latex], this equation reduces to [latex]a^x=e^{x \\log_e a}=e^{x \\ln a}[\/latex].<\/li>\r\n \t<li>[latex]\\log_a x=\\frac{\\log_b x}{\\log_b a}[\/latex] for any real number [latex]x&gt;0[\/latex].\r\nIf [latex]b=e[\/latex], this equation reduces to [latex]\\log_a x=\\frac{\\ln x}{\\ln a}[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1170572219407\" class=\"bc-section section\">\r\n<h2>Proof<\/h2>\r\n<p id=\"fs-id1170572219413\">For the first change-of-base formula, we begin by making use of the power property of logarithmic functions. We know that for any base [latex]b&gt;0, \\, b\\ne 1, \\, \\log_b (a^x)=x \\log_b a[\/latex]. Therefore,<\/p>\r\n\r\n<div id=\"fs-id1170572219472\" class=\"equation unnumbered\">[latex]b^{\\log_b(a^x)}=b^{x \\log_b a}[\/latex].<\/div>\r\n<p id=\"fs-id1170572219523\">In addition, we know that [latex]b^x[\/latex] and [latex]\\log_b (x)[\/latex] are inverse functions. Therefore,<\/p>\r\n\r\n<div id=\"fs-id1170572219556\" class=\"equation unnumbered\">[latex]b^{\\log_b (a^x)}=a^x[\/latex].<\/div>\r\n<p id=\"fs-id1170572309868\">Combining these last two equalities, we conclude that [latex]a^x=b^{x \\log_b a}[\/latex].<\/p>\r\n<p id=\"fs-id1170572309898\">To prove the second property, we show that<\/p>\r\n\r\n<div id=\"fs-id1170572309902\" class=\"equation unnumbered\">[latex](\\log_b a)\u00b7(\\log_a x)=\\log_b x[\/latex].<\/div>\r\n<p id=\"fs-id1170572309957\">Let [latex]u=\\log_b a, \\, v=\\log_a x[\/latex], and [latex]w=\\log_b x[\/latex]. We will show that [latex]u\u00b7v=w[\/latex]. By the definition of logarithmic functions, we know that [latex]b^u=a, \\, a^v=x[\/latex], and [latex]b^w=x[\/latex]. From the previous equations, we see that<\/p>\r\n\r\n<div id=\"fs-id1170572434870\" class=\"equation unnumbered\">[latex]b^{uv}=(b^u)^v=a^v=x=b^w[\/latex].<\/div>\r\n<p id=\"fs-id1170572434927\">Therefore, [latex]b^{uv}=b^w[\/latex]. Since exponential functions are one-to-one, we can conclude that [latex]u\u00b7v=w[\/latex].<\/p>\r\n<p id=\"fs-id1170572434967\">\u25a1<\/p>\r\n\r\n<div id=\"fs-id1170572434970\" class=\"textbox examples\">\r\n<h3>Changing Bases<\/h3>\r\n<div id=\"fs-id1170572434975\" class=\"exercise\">\r\n<div id=\"fs-id1170572434977\" class=\"textbox\">\r\n<p id=\"fs-id1170572434979\">Use a calculating utility to evaluate [latex]\\log_3 7[\/latex] with the change-of-base formula presented earlier.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572435001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572435001\"]\r\n<p id=\"fs-id1170572435001\">Use the second equation with [latex]a=3[\/latex] and [latex]e=3[\/latex]:<\/p>\r\n<p id=\"fs-id1170572435025\">[latex]\\log_3 7=\\frac{\\ln 7}{\\ln 3} \\approx 1.77124[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572435059\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572435063\" class=\"exercise\">\r\n<div id=\"fs-id1170572435065\" class=\"textbox\">\r\n<p id=\"fs-id1170572435068\">Use the change-of-base formula and a calculating utility to evaluate [latex]\\log_4 6[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572128662\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572128662\"]\r\n<p id=\"fs-id1170572128662\">1.29248<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165043161231\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042853660\">Use the change of base to rewrite this expression in terms of expressions involving the natural logarithm function.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572128672\" class=\"textbox examples\">\r\n<h3>Chapter Opener: The Richter Scale for Earthquakes<\/h3>\r\n<div id=\"fs-id1170572128677\" class=\"exercise\">\r\n<div id=\"fs-id1170572128679\" class=\"textbox\">\r\n<div id=\"CNX_Calc_Figure_01_05_010\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202656\/CNX_Calc_Figure_01_05_013.jpg\" alt=\"A photograph of an earthquake fault.\" width=\"325\" height=\"111\" \/> Figure 6: (credit: modification of work by Robb Hannawacker, NPS)[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1170572128701\">In 1935, Charles Richter developed a scale (now known as the <span class=\"no-emphasis\"><em>Richter scale<\/em><\/span>) to measure the magnitude of an <span class=\"no-emphasis\">earthquake<\/span>. The scale is a base-10 logarithmic scale, and it can be described as follows: Consider one earthquake with magnitude [latex]R_1[\/latex] on the Richter scale and a second earthquake with magnitude [latex]R_2[\/latex] on the Richter scale. Suppose [latex]R_1 &gt; R_2[\/latex], which means the earthquake of magnitude [latex]R_1[\/latex] is stronger, but how much stronger is it than the other earthquake? A way of measuring the intensity of an earthquake is by using a seismograph to measure the amplitude of the earthquake waves. If [latex]A_1[\/latex] is the amplitude measured for the first earthquake and [latex]A_2[\/latex] is the amplitude measured for the second earthquake, then the amplitudes and magnitudes of the two earthquakes satisfy the following equation:<\/p>\r\n\r\n<div id=\"fs-id1170572128784\" class=\"equation unnumbered\">[latex]R_1 - R_2 = \\log_{10}(\\frac{A_1}{A_2})[\/latex].<\/div>\r\n<p id=\"fs-id1170572128836\">Consider an earthquake that measures 8 on the Richter scale and an earthquake that measures 7 on the Richter scale. Then,<\/p>\r\n\r\n<div id=\"fs-id1170572128840\" class=\"equation unnumbered\">[latex]8-7=\\log_{10}(\\frac{A_1}{A_2})[\/latex].<\/div>\r\n<p id=\"fs-id1170572233764\">Therefore,<\/p>\r\n\r\n<div id=\"fs-id1170572233767\" class=\"equation unnumbered\">[latex]\\log_{10}(\\frac{A_1}{A_2})=1[\/latex],<\/div>\r\n<p id=\"fs-id1170572233809\">which implies [latex]A_1 \/ A_2 = 10[\/latex] or [latex]A_1 = 10A_2[\/latex]. Since [latex]A_1[\/latex] is 10 times the size of [latex]A_2[\/latex], we say that the first earthquake is 10 times as intense as the second earthquake. On the other hand, if one earthquake measures 8 on the Richter scale and another measures 6, then the relative intensity of the two earthquakes satisfies the equation<\/p>\r\n\r\n<div id=\"fs-id1170572233874\" class=\"equation unnumbered\">[latex]\\log_{10}(\\frac{A_1}{A_2})=8-6=2[\/latex].<\/div>\r\n<p id=\"fs-id1170572233924\">Therefore, [latex]A_1=100A_2[\/latex]. That is, the first earthquake is 100 times more intense than the second earthquake.<\/p>\r\n<p id=\"fs-id1170572477962\">How can we use logarithmic functions to compare the relative severity of the magnitude 9 earthquake in Japan in 2011 with the magnitude 7.3 earthquake in Haiti in 2010?<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572477974\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572477974\"]\r\n<p id=\"fs-id1170572477974\">To compare the Japan and Haiti earthquakes, we can use an equation presented earlier:<\/p>\r\n<p id=\"fs-id1170572477977\">[latex]9-7.3=\\log_{10}(\\frac{A_1}{A_2})[\/latex].<\/p>\r\n<p id=\"fs-id1170572478019\">Therefore, [latex]A_1 \/ A_2=10^{1.7}[\/latex], and we conclude that the earthquake in Japan was approximately 50 times more intense than the earthquake in Haiti.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572478059\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572478063\" class=\"exercise\">\r\n<div id=\"fs-id1170572478065\" class=\"textbox\">\r\n<p id=\"fs-id1170572478067\">Compare the relative severity of a magnitude 8.4 earthquake with a magnitude 7.4 earthquake.<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<p id=\"fs-id1170572478135\">The magnitude 8.4 earthquake is roughly 10 times as severe as the magnitude 7.4 earthquake.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165039564589\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165039564597\">[latex]R_1-R_2=\\log_{10}(A_1 \/ A_2)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572478160\" class=\"bc-section section\">\r\n<h1>Hyperbolic Functions<\/h1>\r\n<p id=\"fs-id1170572467918\">The <strong>hyperbolic functions<\/strong> are defined in terms of certain combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]. These functions arise naturally in various engineering and physics applications, including the study of water waves and vibrations of elastic membranes. Another common use for a hyperbolic function is the representation of a hanging chain or cable, also known as a catenary (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_006\">(Figure)<\/a>). If we introduce a coordinate system so that the low point of the chain lies along the [latex]y[\/latex]-axis, we can describe the height of the chain in terms of a hyperbolic function. First, we define the hyperbolic functions.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_01_05_006\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202700\/CNX_Calc_Figure_01_05_009.jpg\" alt=\"A photograph of a spider web collecting dew drops.\" width=\"488\" height=\"403\" \/> Figure 7. The shape of a strand of silk in a spider\u2019s web can be described in terms of a hyperbolic function. The same shape applies to a chain or cable hanging from two supports with only its own weight. (credit: \u201cMtpaley\u201d, Wikimedia Commons)[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1170572467985\" class=\"textbox key-takeaways\">\r\n<div class=\"title\">\r\n<h3>Definition<\/h3>\r\n<\/div>\r\n<p id=\"fs-id1170572467989\"><strong>Hyperbolic cosine<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572467996\" class=\"equation unnumbered\">[latex]\\cosh x=\\large \\frac{e^x+e^{\u2212x}}{2}[\/latex]<\/div>\r\n<p id=\"fs-id1170572468030\"><strong>Hyperbolic sine<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572468036\" class=\"equation unnumbered\">[latex]\\sinh x=\\large \\frac{e^x-e^{\u2212x}}{2}[\/latex]<\/div>\r\n<p id=\"fs-id1170572468070\"><strong>Hyperbolic tangent<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572468077\" class=\"equation unnumbered\">[latex]\\tanh x=\\large \\frac{\\sinh x}{\\cosh x} \\normalsize = \\large \\frac{e^x-e^{\u2212x}}{e^x+e^{\u2212x}}[\/latex]<\/div>\r\n<p id=\"fs-id1170572431436\"><strong>Hyperbolic cosecant<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572431443\" class=\"equation unnumbered\">[latex]\\text{csch} \\, x=\\large \\frac{1}{\\sinh x} \\normalsize = \\large \\frac{2}{e^x-e^{\u2212x}}[\/latex]<\/div>\r\n<p id=\"fs-id1170572431488\"><strong>Hyperbolic secant<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572431494\" class=\"equation unnumbered\">[latex]\\text{sech} \\, x=\\large \\frac{1}{\\cosh x} \\normalsize = \\large \\frac{2}{e^x+e^{\u2212x}}[\/latex]<\/div>\r\n<p id=\"fs-id1170572431539\"><strong>Hyperbolic cotangent<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572431545\" class=\"equation unnumbered\">[latex]\\coth x=\\large \\frac{\\cosh x}{\\sinh x} \\normalsize = \\large \\frac{e^x+e^{\u2212x}}{e^x-e^{\u2212x}}[\/latex]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572431609\">The name <em>cosh<\/em> rhymes with \u201cgosh,\u201d whereas the name <em>sinh<\/em> is pronounced \u201ccinch.\u201d <em>Tanh<\/em>, <em>sech<\/em>, <em>csch<\/em>, and <em>coth<\/em> are pronounced \u201ctanch,\u201d \u201cseech,\u201d \u201ccoseech,\u201d and \u201ccotanch,\u201d respectively.<\/p>\r\n<p id=\"fs-id1170572234293\">Using the definition of [latex]\\cosh(x)[\/latex] and principles of physics, it can be shown that the height of a hanging chain, such as the one in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_006\">(Figure)<\/a>, can be described by the function [latex]h(x)=a \\cosh(x\/a)+c[\/latex] for certain constants [latex]a[\/latex] and [latex]c[\/latex].<\/p>\r\n<p id=\"fs-id1170572234364\">But why are these functions called <em>hyperbolic functions<\/em>? To answer this question, consider the quantity [latex]\\cosh^2 t-\\sinh^2 t[\/latex]. Using the definition of [latex]\\cosh[\/latex] and [latex]\\sinh[\/latex], we see that<\/p>\r\n\r\n<div id=\"fs-id1170572234411\" class=\"equation unnumbered\">[latex]\\cosh^2 t-\\sinh^2 t=\\large \\frac{e^{2t}+2+e^{-2t}}{4}-\\frac{e^{2t}-2+e^{-2t}}{4} \\normalsize =1[\/latex].<\/div>\r\n<p id=\"fs-id1170572171610\">This identity is the analog of the trigonometric identity [latex]\\cos^2 t+\\sin^2 t=1[\/latex]. Here, given a value [latex]t[\/latex], the point [latex](x,y)=(\\cosh t,\\sinh t)[\/latex] lies on the unit hyperbola [latex]x^2-y^2=1[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_007\">(Figure)<\/a>).<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_01_05_007\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202702\/CNX_Calc_Figure_01_05_007.jpg\" alt=\"An image of a graph. The x axis runs from -1 to 3 and the y axis runs from -3 to 3. The graph is of the relation \u201c(x squared) - (y squared) -1\u201d. The left most point of the relation is at the x intercept, which is at the point (1, 0). From this point the relation both increases and decreases in curves as x increases. This relation is known as a hyperbola and it resembles a sideways \u201cU\u201d shape. There is a point plotted on the graph of the relation labeled \u201c(cosh(1), sinh(1))\u201d, which is at the approximate point (1.5, 1.2).\" width=\"325\" height=\"275\" \/> Figure 8.The unit hyperbola [latex]\\cosh^2 t-\\sinh^2 t=1[\/latex].[\/caption]<\/div>\r\n<div id=\"fs-id1170572171758\" class=\"bc-section section\">\r\n<h2>Graphs of Hyperbolic Functions<\/h2>\r\n<p id=\"fs-id1170572171763\">To graph [latex]\\cosh x[\/latex] and [latex]\\sinh x[\/latex], we make use of the fact that both functions approach [latex](1\/2)e^x[\/latex] as [latex]x \\to \\infty[\/latex], since [latex]e^{\u2212x} \\to 0[\/latex] as [latex]x \\to \\infty[\/latex]. As [latex]x \\to \u2212\\infty, \\, \\cosh x[\/latex] approaches [latex]1\/2e^{\u2212x}[\/latex], whereas [latex]\\sinh x[\/latex] approaches [latex]-1\/2e^{\u2212x}[\/latex]. Therefore, using the graphs of [latex]1\/2e^x, \\, 1\/2e^{\u2212x}[\/latex], and [latex]\u22121\/2e^{\u2212x}[\/latex] as guides, we graph [latex]\\cosh x[\/latex] and [latex]\\sinh x[\/latex]. To graph [latex]\\tanh x[\/latex], we use the fact that [latex]\\tanh(0)=1, \\, -1&lt;\\tanh(x)&lt;1[\/latex] for all [latex]x, \\, \\tanh x \\to 1[\/latex] as [latex]x \\to \\infty[\/latex], and [latex]\\tanh x \\to \u22121[\/latex] as [latex]x \\to \u2212\\infty[\/latex]. The graphs of the other three hyperbolic functions can be sketched using the graphs of [latex]\\cosh x, \\, \\sinh x[\/latex], and [latex]\\tanh x[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_008\">(Figure)<\/a>).<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_01_05_008\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"573\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202706\/CNX_Calc_Figure_01_05_011.jpg\" alt=\"An image of six graphs. Each graph has an x axis that runs from -3 to 3 and a y axis that runs from -4 to 4. The first graph is of the function \u201cy = cosh(x)\u201d, which is a hyperbola. The function decreases until it hits the point (0, 1), where it begins to increase. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = (1\/2)(e to power of -x)\u201d, a decreasing curved function and the second of these functions is \u201cy = (1\/2)(e to power of x)\u201d, an increasing curved function. The function \u201cy = cosh(x)\u201d is always above these two functions without ever touching them. The second graph is of the function \u201cy = sinh(x)\u201d, which is an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = (1\/2)(e to power of x)\u201d, an increasing curved function and the second of these functions is \u201cy = -(1\/2)(e to power of -x)\u201d, an increasing curved function that approaches the x axis without touching it. The function \u201cy = sinh(x)\u201d is always between these two functions without ever touching them. The third graph is of the function \u201cy = sech(x)\u201d, which increases until the point (0, 1), where it begins to decrease. The graph of the function has a hump. The fourth graph is of the function \u201cy = csch(x)\u201d. On the left side of the y axis, the function starts slightly below the x axis and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the x axis, which it never touches. The fifth graph is of the function \u201cy = tanh(x)\u201d, an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = 1\u201d, a horizontal line function and the second of these functions is \u201cy = -1\u201d, another horizontal line function. The function \u201cy = tanh(x)\u201d is always between these two functions without ever touching them. The sixth graph is of the function \u201cy = coth(x)\u201d. On the left side of the y axis, the function starts slightly below the boundary line \u201cy = 1\u201d and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the boundary line \u201cy = -1\u201d, which it never touches.\" width=\"573\" height=\"929\" \/> Figure 9. The hyperbolic functions involve combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex].[\/caption]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572433248\" class=\"bc-section section\">\r\n<h2>Identities Involving Hyperbolic Functions<\/h2>\r\n<p id=\"fs-id1170572433254\">The identity [latex]\\cosh^2 t-\\sinh^2 t[\/latex], shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_007\">(Figure)<\/a>, is one of several identities involving the hyperbolic functions, some of which are listed next. The first four properties follow easily from the definitions of hyperbolic sine and hyperbolic cosine. Except for some differences in signs, most of these properties are analogous to identities for trigonometric functions.<\/p>\r\n\r\n<div id=\"fs-id1170572433288\" class=\"textbox key-takeaways\">\r\n<h3>Rule: Identities Involving Hyperbolic Functions<\/h3>\r\n<ol id=\"fs-id1170572433294\">\r\n \t<li>[latex]\\cosh(\u2212x)=\\cosh x[\/latex]<\/li>\r\n \t<li>[latex]\\sinh(\u2212x)=\u2212\\sinh x[\/latex]<\/li>\r\n \t<li>[latex]\\cosh x+\\sinh x=e^x[\/latex]<\/li>\r\n \t<li>[latex]\\cosh x-\\sinh x=e^{\u2212x}[\/latex]<\/li>\r\n \t<li>[latex]\\cosh^2 x-\\sinh^2 x=1[\/latex]<\/li>\r\n \t<li>[latex]1-\\tanh^2 x=\\text{sech}^2 x[\/latex]<\/li>\r\n \t<li>[latex]\\coth^2 x-1=\\text{csch}^2 x[\/latex]<\/li>\r\n \t<li>[latex]\\sinh(x \\pm y)=\\sinh x \\cosh y \\pm \\cosh x \\sinh y[\/latex]<\/li>\r\n \t<li>[latex]\\cosh (x \\pm y)=\\cosh x \\cosh y \\pm \\sinh x \\sinh y[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1170572443393\" class=\"textbox examples\">\r\n<h3>Evaluating Hyperbolic Functions<\/h3>\r\n<div id=\"fs-id1170572443399\" class=\"exercise\">\r\n<div id=\"fs-id1170572443401\" class=\"textbox\">\r\n<ol id=\"fs-id1170572443403\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Simplify [latex]\\sinh(5 \\ln x)[\/latex].<\/li>\r\n \t<li>If [latex]\\sinh x=3\/4[\/latex], find the values of the remaining five hyperbolic functions.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572443462\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572443462\"]\r\n<ol id=\"fs-id1170572443462\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Using the definition of the [latex]\\sinh[\/latex] function, we write\r\n<div id=\"fs-id1170570995857\" class=\"equation unnumbered\">[latex]\\sinh(5 \\ln x)=\\large \\frac{e^{5 \\ln x}-e^{-5 \\ln x}}{2} \\normalsize = \\large \\frac{e^{\\ln(x^5)}-e^{\\ln(x^{-5})}}{2} \\normalsize =\\large \\frac{x^5-x^{-5}}{2}[\/latex].<\/div><\/li>\r\n \t<li>Using the identity [latex]\\cosh^2 x-\\sinh^2 x=1[\/latex], we see that\r\n<div id=\"fs-id1170573388429\" class=\"equation unnumbered\">[latex]\\cosh^2 x=1+\\big(\\frac{3}{4}\\big)^2=\\frac{25}{16}[\/latex].<\/div>\r\nSince [latex]\\cosh x \\ge 1[\/latex] for all [latex]x[\/latex], we must have [latex]\\cosh x=5\/4[\/latex]. Then, using the definitions for the other hyperbolic functions, we conclude that [latex]\\tanh x=3\/5, \\, \\text{csch} \\, x=4\/3, \\, \\text{sech} \\, x=4\/5[\/latex], and [latex]\\coth x=5\/3[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572548314\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572548318\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1170572549916\">Simplify [latex]\\cosh(2 \\ln x)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572549946\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572549946\"]\r\n<p id=\"fs-id1170572549946\">[latex](x^2+x^{-2})\/2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165039563328\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165039563336\">Use the definition of the cosh function and the power property of logarithm functions.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572549981\" class=\"bc-section section\">\r\n<h2>Inverse Hyperbolic Functions<\/h2>\r\n<p id=\"fs-id1170572549987\">From the graphs of the hyperbolic functions, we see that all of them are one-to-one except [latex]\\cosh x[\/latex] and [latex]\\text{sech} \\, x[\/latex]. If we restrict the domains of these two functions to the interval [latex][0,\\infty)[\/latex], then all the hyperbolic functions are one-to-one, and we can define the inverse hyperbolic functions. Since the hyperbolic functions themselves involve exponential functions, the inverse hyperbolic functions involve logarithmic functions.<\/p>\r\n\r\n<div id=\"fs-id1170572550032\" class=\"textbox key-takeaways\">\r\n<div class=\"title\">\r\n<h3>Definition<\/h3>\r\n<\/div>\r\n<p id=\"fs-id1170572550036\"><strong>Inverse Hyperbolic Functions<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572550043\" class=\"equation unnumbered\">[latex]\\begin{array}{cccc}\\sinh^{-1} x=\\text{arcsinh } x=\\ln(x+\\sqrt{x^2+1})\\hfill &amp; &amp; &amp; \\cosh^{-1} x=\\text{arccosh } x=\\ln(x+\\sqrt{x^2-1})\\hfill \\\\ \\tanh^{-1} x=\\text{arctanh } x=\\frac{1}{2}\\ln\\big(\\frac{1+x}{1-x}\\big)\\hfill &amp; &amp; &amp; \\coth^{-1} x=\\text{arccot } x=\\frac{1}{2}\\ln\\big(\\frac{x+1}{x-1}\\big)\\hfill \\\\ \\text{sech}^{-1} x=\\text{arcsech } x=\\ln\\big(\\frac{1+\\sqrt{1-x^2}}{x}\\big)\\hfill &amp; &amp; &amp; \\text{csch}^{-1} x=\\text{arccsch } x=\\ln\\big(\\frac{1}{x}+\\frac{\\sqrt{1+x^2}}{|x|}\\big)\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572235136\">Let\u2019s look at how to derive the first equation. The others follow similarly. Suppose [latex]y=\\sinh^{-1} x[\/latex]. Then, [latex]x=\\sinh y[\/latex] and, by the definition of the hyperbolic sine function, [latex]x=\\frac{e^y-e^{\u2212y}}{2}[\/latex]. Therefore,<\/p>\r\n\r\n<div id=\"fs-id1170572235202\" class=\"equation unnumbered\">[latex]e^y-2x-e^{\u2212y}=0[\/latex].<\/div>\r\nMultiplying this equation by [latex]e^y[\/latex], we obtain\r\n<div class=\"equation unnumbered\">[latex]e^{2y}-2xe^y-1=0[\/latex].<\/div>\r\n<p id=\"fs-id1170572235286\">This can be solved like a quadratic equation, with the solution<\/p>\r\n\r\n<div id=\"fs-id1170572235290\" class=\"equation unnumbered\">[latex]e^y=\\large \\frac{2x \\pm \\sqrt{4x^2+4}}{2} \\normalsize =x \\pm \\sqrt{x^2+1}[\/latex].<\/div>\r\n<p id=\"fs-id1170572482176\">Since [latex]e^y&gt;0[\/latex], the only solution is the one with the positive sign. Applying the natural logarithm to both sides of the equation, we conclude that<\/p>\r\n\r\n<div id=\"fs-id1170572482196\" class=\"equation unnumbered\">[latex]y=\\ln(x+\\sqrt{x^2+1})[\/latex].<\/div>\r\n<div id=\"fs-id1170572482234\" class=\"textbox examples\">\r\n<h3>Evaluating Inverse Hyperbolic Functions<\/h3>\r\n<div id=\"fs-id1170572482240\" class=\"exercise\">\r\n<div id=\"fs-id1170572482242\" class=\"textbox\">\r\n<p id=\"fs-id1170572482244\">Evaluate each of the following expressions.<\/p>\r\n\r\n<div id=\"fs-id1170572482247\" class=\"equation unnumbered\">[latex]\\sinh^{-1}(2)[\/latex]<\/div>\r\n<div id=\"fs-id1170572482270\" class=\"equation unnumbered\">[latex]\\tanh^{-1}(1\/4)[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572482306\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572482306\"]\r\n<p id=\"fs-id1170572482306\">[latex]\\sinh^{-1}(2)=\\ln(2+\\sqrt{2^2+1})=\\ln(2+\\sqrt{5}) \\approx 1.4436[\/latex]<\/p>\r\n<p id=\"fs-id1170572176029\">[latex]\\tanh^{-1}(1\/4)=\\frac{1}{2}\\ln(\\frac{1+1\/4}{1-1\/4})=\\frac{1}{2}\\ln(\\frac{5\/4}{3\/4})=\\frac{1}{2}\\ln(\\frac{5}{3}) \\approx 0.2554[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572176157\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572176161\" class=\"exercise\">\r\n<div id=\"fs-id1170572176163\" class=\"textbox\">\r\n<p id=\"fs-id1170572176165\">Evaluate [latex]\\tanh^{-1}(1\/2)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572176215\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572176215\"]\r\n<p id=\"fs-id1170572176215\">[latex]\\frac{1}{2}\\ln(3) \\approx 0.5493[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165039559588\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165039559595\">Use the definition of [latex]\\tanh^{-1} x[\/latex] and simplify.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572176246\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1170572216270\">\r\n \t<li>The exponential function [latex]y=b^x[\/latex] is increasing if [latex]b&gt;1[\/latex] and decreasing if [latex]0&lt;b&lt;1[\/latex]. Its domain is [latex](\u2212\\infty ,\\infty)[\/latex] and its range is [latex](0,\\infty)[\/latex].<\/li>\r\n \t<li>The logarithmic function [latex]y=\\log_b(x)[\/latex] is the inverse of [latex]y=b^x[\/latex]. Its domain is [latex](0,\\infty)[\/latex] and its range is [latex](\u2212\\infty,\\infty)[\/latex].<\/li>\r\n \t<li>The natural exponential function is [latex]y=e^x[\/latex] and the natural logarithmic function is [latex]y=\\ln x=\\log_e x[\/latex].<\/li>\r\n \t<li>Given an exponential function or logarithmic function in base [latex]a[\/latex], we can make a change of base to convert this function to any base [latex]b&gt;0, \\, b \\ne 1[\/latex]. We typically convert to base [latex]e[\/latex].<\/li>\r\n \t<li>The hyperbolic functions involve combinations of the exponential functions [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]. As a result, the inverse hyperbolic functions involve the natural logarithm.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1170572545884\" class=\"textbox exercises\">\r\n<p id=\"fs-id1170572545888\">For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal.<\/p>\r\n\r\n<div id=\"fs-id1170572545893\" class=\"exercise\">\r\n<div id=\"fs-id1170572545895\" class=\"textbox\">\r\n<p id=\"fs-id1170572545897\"><strong>1.\u00a0<\/strong>[latex]f(x)=5^x[\/latex] a. [latex]x=3[\/latex] b. [latex]x=\\frac{1}{2}[\/latex] c. [latex]x=\\sqrt{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572545955\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572545955\"]\r\n<p id=\"fs-id1170572545955\">a. 125 b. 2.24 c. 9.74<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572545960\" class=\"exercise\">\r\n<div id=\"fs-id1170572545962\" class=\"textbox\">\r\n<p id=\"fs-id1170572545965\"><strong>2.\u00a0<\/strong>[latex]f(x)=(0.3)^x[\/latex] a. [latex]x=-1[\/latex] b. [latex]x=4[\/latex] c. [latex]x=-1.5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572546032\" class=\"exercise\">\r\n<div id=\"fs-id1170572546034\" class=\"textbox\">\r\n<p id=\"fs-id1170572546036\"><strong>3.\u00a0<\/strong>[latex]f(x)=10^x[\/latex] a. [latex]x=-2[\/latex] b. [latex]x=4[\/latex] c. [latex]x=\\frac{5}{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572358150\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572358150\"]\r\n<p id=\"fs-id1170572358150\">a. 0.01 b. 10,000 c. 46.42<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572358156\" class=\"exercise\">\r\n<div id=\"fs-id1170572358158\" class=\"textbox\">\r\n<p id=\"fs-id1170572358160\"><strong>4.\u00a0<\/strong>[latex]f(x)=e^x[\/latex] a. [latex]x=2[\/latex] b. [latex]x=-3.2[\/latex] c. [latex]x=\\pi [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572358219\">For the following exercises, match the exponential equation to the correct graph.<\/p>\r\n\r\n<ol id=\"fs-id1170572358222\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]y=4^{\u2212x}[\/latex]<\/li>\r\n \t<li>[latex]y=3^{x-1}[\/latex]<\/li>\r\n \t<li>[latex]y=2^{x+1}[\/latex]<\/li>\r\n \t<li>[latex]y=(\\frac{1}{2})^x+2[\/latex]<\/li>\r\n \t<li>[latex]y=\u22123^{\u2212x}[\/latex]<\/li>\r\n \t<li>[latex]y=1-5^x[\/latex]<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1170572358351\" class=\"exercise\">\r\n<div id=\"fs-id1170572358353\" class=\"textbox\"><span id=\"fs-id1170572358359\"><strong>5.\u00a0<\/strong><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202709\/CNX_Calc_Figure_01_05_201.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -2 to 8. The graph is of a decreasing curved function. The function decreases until it approaches the line \u201cy = 2\u201d, but never touches this line. The y intercept is at the point (0, 3) and there is no x intercept.\" \/><\/span><\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572553700\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572553700\"]\r\n<p id=\"fs-id1170572553700\">d<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572553705\" class=\"exercise\">\r\n<div id=\"fs-id1170572553707\" class=\"textbox\"><span id=\"fs-id1170572553713\"><strong>6.\u00a0<\/strong><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202713\/CNX_Calc_Figure_01_05_202.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -9 to 2. The graph is of a function that starts slightly below the line \u201cy = 1\u201d and begins decreasing rapidly in a curve. The x intercept and y intercept are both at the origin.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572553732\" class=\"exercise\">\r\n<div id=\"fs-id1170572553734\" class=\"textbox\"><span id=\"fs-id1170572553737\"><strong>7.\u00a0<\/strong><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202716\/CNX_Calc_Figure_01_05_203.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that starts slightly above the x axis and begins increasing rapidly. There is no x intercept and the y intercept is at the point (0, (1\/3)). Another point of the graph is at (1, 1).\" \/><\/span><\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572553754\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572553754\"]\r\n<p id=\"fs-id1170572553754\">b<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572553760\" class=\"exercise\">\r\n<div id=\"fs-id1170572553762\" class=\"textbox\"><span id=\"fs-id1170572553764\"><strong>8.\u00a0<\/strong><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202719\/CNX_Calc_Figure_01_05_204.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved decreasing function that decreases until it comes close the x axis without touching it. There is no x intercept and the y intercept is at the point (0, 1). Another point of the graph is at (-1, 4).\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572553787\" class=\"exercise\">\r\n<div id=\"fs-id1170572553789\" class=\"textbox\"><span id=\"fs-id1170572553791\"><strong>9.\u00a0<\/strong><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202722\/CNX_Calc_Figure_01_05_205.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that increases until it comes close the x axis without touching it. There is no x intercept and the y intercept is at the point (0, -1). Another point of the graph is at (-1, -3).\" \/><\/span><\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572553809\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572553809\"]\r\n<p id=\"fs-id1170572553809\">e<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572553814\" class=\"exercise\">\r\n<div id=\"fs-id1170572553816\" class=\"textbox\"><span id=\"fs-id1170572553818\"><strong>10.<\/strong><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202725\/CNX_Calc_Figure_01_05_206.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that starts slightly above the x axis and begins increasing rapidly. There is no x intercept and the y intercept is at the point (0, 2). Another point of the graph is at (-1, 1).\" \/><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572553841\">For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote.<\/p>\r\n\r\n<div id=\"fs-id1170572553846\" class=\"exercise\">\r\n<div id=\"fs-id1170572553848\" class=\"textbox\">\r\n<p id=\"fs-id1170572553850\"><strong>11.\u00a0<\/strong>[latex]f(x)=e^x+2[\/latex]<\/p>\r\n<span id=\"fs-id1170572553879\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202729\/CNX_Calc_Figure_01_05_207.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that starts slightly above the line \u201cy = 2\u201d and begins increasing rapidly. There is no x intercept and the y intercept is at the point (0, 3).\" \/><\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572553893\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572553893\"]\r\n<p id=\"fs-id1170572553893\">Domain: all real numbers, Range: [latex](2,\\infty)[\/latex], Horizontal asymptote at [latex]y=2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572553923\" class=\"exercise\">\r\n<div id=\"fs-id1170572553925\" class=\"textbox\">\r\n<p id=\"fs-id1170572553927\"><strong>12.\u00a0<\/strong>[latex]f(x)=\u22122^x[\/latex]<\/p>\r\n<span id=\"fs-id1170572242925\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202731\/CNX_Calc_Figure_01_05_208.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a function that starts slightly below the x axis and begins decreasing rapidly in a curve. There is no x intercept and y intercept is at the point (0, -1).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572242973\" class=\"exercise\">\r\n<div id=\"fs-id1170572242975\" class=\"textbox\">\r\n<p id=\"fs-id1170572242977\"><strong>13.\u00a0<\/strong>[latex]f(x)=3^{x+1}[\/latex]<\/p>\r\n<span id=\"fs-id1170572243004\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202734\/CNX_Calc_Figure_01_05_209.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that starts slightly above the x axis and begins increasing rapidly. There is no x intercept and the y intercept is at the point (0, 3). Another point of the graph is at (-1, 1).\" \/><\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572243022\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572243022\"]\r\n<p id=\"fs-id1170572243022\">Domain: all real numbers, Range: [latex](0,\\infty)[\/latex], Horizontal asymptote at [latex]y=0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572243051\" class=\"exercise\">\r\n<div id=\"fs-id1170572243053\" class=\"textbox\">\r\n<p id=\"fs-id1170572243055\"><strong>14.\u00a0<\/strong>[latex]f(x)=4^x-1[\/latex]<\/p>\r\n<span id=\"fs-id1170572243084\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202738\/CNX_Calc_Figure_01_05_210.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that starts slightly above the line \u201cy = -1\u201d and begins increasing rapidly. There x intercept and the y intercept are both at the origin. Another point of the graph is at (1, 3).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572243129\" class=\"exercise\">\r\n<div id=\"fs-id1170572243131\" class=\"textbox\">\r\n<p id=\"fs-id1170572243133\"><strong>15.\u00a0<\/strong>[latex]f(x)=1-2^{\u2212x}[\/latex]<\/p>\r\n<span id=\"fs-id1170572163713\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202741\/CNX_Calc_Figure_01_05_211.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that increases until it comes close the line \u201cy = 1\u201d without touching it. There x intercept and the y intercept are both at the origin. Another point of the graph is at (-1, -1).\" \/><\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572163728\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572163728\"]\r\n<p id=\"fs-id1170572163728\">Domain: all real numbers, Range: [latex](\u2212\\infty ,1)[\/latex], Horizontal asymptote at [latex]y=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572163759\" class=\"exercise\">\r\n<div id=\"fs-id1170572163761\" class=\"textbox\">\r\n<p id=\"fs-id1170572163763\"><strong>16.\u00a0<\/strong>[latex]f(x)=5^{x+1}+2[\/latex]<\/p>\r\n<span id=\"fs-id1170572163798\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202744\/CNX_Calc_Figure_01_05_212.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -1 to 9. The graph is of a curved increasing function that starts slightly above the line \u201cy = 2\u201d and begins increasing rapidly. There is no x intercept and the y intercept is at the point (0, 7). Another point of the graph is at (-1, 3).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572163842\" class=\"exercise\">\r\n<div id=\"fs-id1170572163844\" class=\"textbox\">\r\n<p id=\"fs-id1170572163846\"><strong>17.\u00a0<\/strong>[latex]f(x)=e^{\u2212x}-1[\/latex]<\/p>\r\n<span id=\"fs-id1170572163879\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202748\/CNX_Calc_Figure_01_05_213.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved decreasing function that decreases until it comes close the line \u201cy = -1\u201d without touching it. There x intercept and the y intercept are both at the origin. There is an approximate point on the graph at (-1, 1.7).\" \/><\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572163893\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572163893\"]\r\n<p id=\"fs-id1170572163893\">Domain: all real numbers, Range: [latex](-1,\\infty )[\/latex], Horizontal asymptote at [latex]y=-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572163923\">For the following exercises, write the equation in equivalent exponential form.<\/p>\r\n\r\n<div id=\"fs-id1170572163926\" class=\"exercise\">\r\n<div id=\"fs-id1170572163928\" class=\"textbox\">\r\n<p id=\"fs-id1170572163930\"><strong>18.\u00a0<\/strong>[latex]\\log_3 81=4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572218452\" class=\"exercise\">\r\n<div id=\"fs-id1170572218455\" class=\"textbox\">\r\n<p id=\"fs-id1170572218457\"><strong>19.\u00a0<\/strong>[latex]\\log_8 2=\\frac{1}{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572218480\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572218480\"]\r\n<p id=\"fs-id1170572218480\">[latex]8^{1\/3}=2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572218502\" class=\"exercise\">\r\n<div id=\"fs-id1170572218504\" class=\"textbox\">\r\n<p id=\"fs-id1170572218506\"><strong>20.\u00a0<\/strong>[latex]\\log_5 1=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572218542\" class=\"exercise\">\r\n<div id=\"fs-id1170572218544\" class=\"textbox\">\r\n<p id=\"fs-id1170572218546\"><strong>21.\u00a0<\/strong>[latex]\\log_5 25=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572218567\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572218567\"]\r\n<p id=\"fs-id1170572218567\">[latex]5^2=25[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572218583\" class=\"exercise\">\r\n<div id=\"fs-id1170572218585\" class=\"textbox\">\r\n<p id=\"fs-id1170572218587\"><strong>22.\u00a0<\/strong>[latex]\\log 0.1=-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572218621\" class=\"exercise\">\r\n<div id=\"fs-id1170572218623\" class=\"textbox\">\r\n<p id=\"fs-id1170572218626\"><strong>23.\u00a0<\/strong>[latex]\\ln(\\frac{1}{e^3})=-3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572218656\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572218656\"]\r\n<p id=\"fs-id1170572218656\">[latex]e^{-3}=\\frac{1}{e^3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572168644\" class=\"exercise\">\r\n<div id=\"fs-id1170572168646\" class=\"textbox\">\r\n<p id=\"fs-id1170572168649\"><strong>24.\u00a0<\/strong>[latex]\\log_9 3=0.5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572168686\" class=\"exercise\">\r\n<div id=\"fs-id1170572168688\" class=\"textbox\">\r\n<p id=\"fs-id1170572168690\"><strong>25.\u00a0<\/strong>[latex]\\ln 1=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572168706\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572168706\"]\r\n<p id=\"fs-id1170572168706\">[latex]e^0=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572168722\">For the following exercises, write the equation in equivalent logarithmic form.<\/p>\r\n\r\n<div id=\"fs-id1170572168725\" class=\"exercise\">\r\n<div id=\"fs-id1170572168728\" class=\"textbox\">\r\n<p id=\"fs-id1170572168730\"><strong>26.\u00a0<\/strong>[latex]2^3=8[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572168766\" class=\"exercise\">\r\n<div id=\"fs-id1170572168768\" class=\"textbox\">\r\n<p id=\"fs-id1170572168770\"><strong>27.\u00a0<\/strong>[latex]4^{-2}=\\frac{1}{16}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572168793\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572168793\"]\r\n<p id=\"fs-id1170572168793\">[latex]\\log_4(\\frac{1}{16})=-2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572168823\" class=\"exercise\">\r\n<div id=\"fs-id1170572168825\" class=\"textbox\">\r\n<p id=\"fs-id1170572168827\"><strong>28.\u00a0<\/strong>[latex]10^2=100[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572168860\" class=\"exercise\">\r\n<div id=\"fs-id1170572168862\" class=\"textbox\">\r\n<p id=\"fs-id1170572168865\"><strong>29.\u00a0<\/strong>[latex]9^0=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572168882\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572168882\"]\r\n<p id=\"fs-id1170572168882\">[latex]\\log_9 1=0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572218179\" class=\"exercise\">\r\n<div id=\"fs-id1170572218181\" class=\"textbox\">\r\n<p id=\"fs-id1170572218184\"><strong>30.\u00a0<\/strong>[latex](\\frac{1}{3})^3=\\frac{1}{27}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572218251\" class=\"exercise\">\r\n<div id=\"fs-id1170572218253\" class=\"textbox\">\r\n<p id=\"fs-id1170572218255\"><strong>31.\u00a0<\/strong>[latex]\\sqrt[3]{64}=4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572218273\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572218273\"]\r\n<p id=\"fs-id1170572218273\">[latex]\\log_{64} 4=\\frac{1}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572218297\" class=\"exercise\">\r\n<div id=\"fs-id1170572218299\" class=\"textbox\">\r\n<p id=\"fs-id1170572218301\"><strong>32.\u00a0<\/strong>[latex]e^x=y[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572218333\" class=\"exercise\">\r\n<div id=\"fs-id1170572218335\" class=\"textbox\">\r\n<p id=\"fs-id1170572218337\"><strong>33.\u00a0<\/strong>[latex]9^y=150[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572218354\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572218354\"]\r\n<p id=\"fs-id1170572218354\">[latex]\\log_9 150=y[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572218373\" class=\"exercise\">\r\n<div id=\"fs-id1170572218376\" class=\"textbox\">\r\n<p id=\"fs-id1170572218378\"><strong>34.\u00a0<\/strong>[latex]b^3=45[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572218414\" class=\"exercise\">\r\n<div id=\"fs-id1170572218416\" class=\"textbox\">\r\n<p id=\"fs-id1170572218418\"><strong>35.\u00a0<\/strong>[latex]4^{-3\/2}=0.125[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572217083\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572217083\"]\r\n<p id=\"fs-id1170572217083\">[latex]\\log_4 0.125=-\\frac{3}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572217107\">For the following exercises, use the given graphs of the logarithmic functions to determine their domain, range, and vertical asymptote.<\/p>\r\n\r\n<div id=\"fs-id1170572217112\" class=\"exercise\">\r\n<div id=\"fs-id1170572217114\" class=\"textbox\">\r\n<p id=\"fs-id1170572217116\"><strong>36.\u00a0<\/strong>[latex]f(x)=3+\\ln x[\/latex]<\/p>\r\n<span id=\"fs-id1170572217140\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202751\/CNX_Calc_Figure_01_05_214.jpg\" alt=\"An image of a graph. The x axis runs from -10 to 10 and the y axis runs from -10 to 10. The graph is of an increasing curved function which starts slightly to the right of the y axis. There is no y intercept and the x intercept is at the approximate point (0.05, 0). Another point on the graph is (1, 3).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572217209\" class=\"exercise\">\r\n<div id=\"fs-id1170572217211\" class=\"textbox\">\r\n<p id=\"fs-id1170572217213\"><strong>37.\u00a0<\/strong>[latex]f(x)=\\ln(x-1)[\/latex]<\/p>\r\n<span id=\"fs-id1170572217249\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202754\/CNX_Calc_Figure_01_05_215.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of an increasing curved function which starts slightly to the right of the vertical line \u201cx = 1\u201d. There is no y intercept and the x intercept is at the approximate point (2, 0).\" \/><\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572217264\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572217264\"]\r\n<p id=\"fs-id1170572217264\">Domain: [latex](1,\\infty ),[\/latex] Range: [latex](\u2212\\infty ,\\infty)[\/latex], Vertical asymptote at [latex]x=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572217315\" class=\"exercise\">\r\n<div id=\"fs-id1170572217317\" class=\"textbox\">\r\n<p id=\"fs-id1170572286631\"><strong>38.\u00a0<\/strong>[latex]f(x)=\\ln(\u2212x)[\/latex]<\/p>\r\n<span id=\"fs-id1170572286662\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202758\/CNX_Calc_Figure_01_05_216.jpg\" alt=\"An image of a graph. The x axis runs from -9 to 1 and the y axis runs from -5 to 5. The graph is a curved decreasing function that approaches the y axis without touching it. There is no y intercept and the x intercept is at (-1, 0).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572286731\" class=\"exercise\">\r\n<div id=\"fs-id1170572286734\" class=\"textbox\">\r\n<p id=\"fs-id1170572286736\"><strong>39.\u00a0<\/strong>[latex]f(x)=1-\\ln x[\/latex]<\/p>\r\n<span id=\"fs-id1170572286760\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202801\/CNX_Calc_Figure_01_05_217.jpg\" alt=\"An image of a graph. The x axis runs from -1 to 9 and the y axis runs from -5 to 5. The graph is of a decreasing curved function which starts slightly to the right of the y axis. There is no y intercept and the x intercept is at the point (e, 0).\" \/><\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572286777\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572286777\"]\r\n<p id=\"fs-id1170572286777\">Domain: [latex](0,\\infty)[\/latex], Range: [latex](\u2212\\infty ,\\infty)[\/latex], Vertical asymptote at [latex]x=0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572286826\" class=\"exercise\">\r\n<div id=\"fs-id1170572286828\" class=\"textbox\">\r\n<p id=\"fs-id1170572286831\"><strong>40.\u00a0<\/strong>[latex]f(x)=\\log x-1[\/latex]<\/p>\r\n<span id=\"fs-id1170572286855\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202804\/CNX_Calc_Figure_01_05_218.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 20 and the y axis runs from -10 to 5. The graph is of an increasing curved function which starts slightly to the right of the y axis. There is no y intercept and the x intercept is at the point (10, 0).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572452755\" class=\"exercise\">\r\n<div id=\"fs-id1170572452757\" class=\"textbox\">\r\n<p id=\"fs-id1170572452759\"><strong>41.\u00a0<\/strong>[latex]f(x)=\\ln(x+1)[\/latex]<\/p>\r\n<span id=\"fs-id1170572452796\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202808\/CNX_Calc_Figure_01_05_219.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of an increasing curved function which starts slightly to the right of the vertical line \u201cx = -1\u201d. There y intercept and the x intercept are both at the origin.\" \/><\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572452810\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572452810\"]\r\n<p id=\"fs-id1170572452810\">Domain: [latex](-1,\\infty)[\/latex], Range: [latex](\u2212\\infty ,\\infty)[\/latex], Vertical asymptote at [latex]x=-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572452859\">For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and\/or product of logarithms.<\/p>\r\n\r\n<div id=\"fs-id1170572452864\" class=\"exercise\">\r\n<div id=\"fs-id1170572452866\" class=\"textbox\">\r\n<p id=\"fs-id1170572452868\"><strong>42.\u00a0<\/strong>[latex]\\log x^4y[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572452904\" class=\"exercise\">\r\n<div id=\"fs-id1170572452906\" class=\"textbox\">\r\n<p id=\"fs-id1170572452909\"><strong>43.\u00a0<\/strong>[latex]\\log_3 \\frac{9a^3}{b}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572452934\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572452934\"]\r\n<p id=\"fs-id1170572452934\">[latex]2+3\\log_3 a-\\log_3 b[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572452966\" class=\"exercise\">\r\n<div id=\"fs-id1170572452968\" class=\"textbox\">\r\n<p id=\"fs-id1170572452970\"><strong>44.\u00a0<\/strong>[latex]\\ln a\\sqrt[3]{b}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572546157\" class=\"exercise\">\r\n<div id=\"fs-id1170572546159\" class=\"textbox\">\r\n<p id=\"fs-id1170572546161\"><strong>45.\u00a0<\/strong>[latex]\\log_5 \\sqrt{125xy^3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572546187\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572546187\"]\r\n<p id=\"fs-id1170572546187\">[latex]\\frac{3}{2}+\\frac{1}{2}\\log_5 x+\\frac{3}{2}\\log_5 y[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572546231\" class=\"exercise\">\r\n<div id=\"fs-id1170572546233\" class=\"textbox\">\r\n<p id=\"fs-id1170572546235\"><strong>46.\u00a0<\/strong>[latex]\\log_4 \\frac{\\sqrt[3]{xy}}{64}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572546304\" class=\"exercise\">\r\n<div id=\"fs-id1170572546306\" class=\"textbox\">\r\n<p id=\"fs-id1170572546308\"><strong>47.\u00a0<\/strong>[latex]\\ln(\\frac{6}{\\sqrt{e^3}})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572546336\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572546336\"]\r\n<p id=\"fs-id1170572546336\">[latex]-\\frac{3}{2}+\\ln 6[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572546356\">For the following exercises, solve the exponential equation exactly.<\/p>\r\n\r\n<div id=\"fs-id1170572546359\" class=\"exercise\">\r\n<div id=\"fs-id1170572546361\" class=\"textbox\">\r\n<p id=\"fs-id1170572546363\"><strong>48.\u00a0<\/strong>[latex]5^x=125[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572479927\" class=\"exercise\">\r\n<div id=\"fs-id1170572479929\" class=\"textbox\">\r\n<p id=\"fs-id1170572479931\"><strong>49.\u00a0<\/strong>[latex]e^{3x}-15=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572479956\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572479956\"]\r\n<p id=\"fs-id1170572479956\">[latex]\\frac{\\ln 15}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572479971\" class=\"exercise\">\r\n<div id=\"fs-id1170572479973\" class=\"textbox\">\r\n<p id=\"fs-id1170572479975\"><strong>50.\u00a0<\/strong>[latex]8^x=4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572480004\" class=\"exercise\">\r\n<div id=\"fs-id1170572480006\" class=\"textbox\">\r\n<p id=\"fs-id1170572480008\"><strong>51.\u00a0<\/strong>[latex]4^{x+1}-32=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572480035\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572480035\"]\r\n<p id=\"fs-id1170572480035\">[latex]\\frac{3}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572480046\" class=\"exercise\">\r\n<div id=\"fs-id1170572480049\" class=\"textbox\">\r\n<p id=\"fs-id1170572480051\"><strong>52.\u00a0<\/strong>[latex]3^{x\/14}=\\frac{1}{10}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572480100\" class=\"exercise\">\r\n<div id=\"fs-id1170572480102\" class=\"textbox\">\r\n<p id=\"fs-id1170572480104\"><strong>53.\u00a0<\/strong>[latex]10^x=7.21[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572480122\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572480122\"]\r\n<p id=\"fs-id1170572480122\">[latex]\\log 7.21[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572480133\" class=\"exercise\">\r\n<div id=\"fs-id1170572480135\" class=\"textbox\">\r\n<p id=\"fs-id1170572480137\"><strong>54.\u00a0<\/strong>[latex]4\u00b72^{3x}-20=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572480186\" class=\"exercise\">\r\n<div id=\"fs-id1170572543576\" class=\"textbox\">\r\n<p id=\"fs-id1170572543578\"><strong>55.\u00a0<\/strong>[latex]7^{3x-2}=11[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572543602\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572543602\"]\r\n<p id=\"fs-id1170572543602\">[latex]\\frac{2}{3}+\\frac{\\log 11}{3\\log 7}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572543630\">For the following exercises, solve the logarithmic equation exactly, if possible.<\/p>\r\n\r\n<div id=\"fs-id1170572543633\" class=\"exercise\">\r\n<div id=\"fs-id1170572543636\" class=\"textbox\">\r\n<p id=\"fs-id1170572543638\"><strong>56.\u00a0<\/strong>[latex]\\log_3 x=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572543671\" class=\"exercise\">\r\n<div id=\"fs-id1170572543673\" class=\"textbox\">\r\n<p id=\"fs-id1170572543675\"><strong>57.\u00a0<\/strong>[latex]\\log_5 x=-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572543696\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572543696\"]\r\n<p id=\"fs-id1170572543696\">[latex]x=\\frac{1}{25}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572543713\" class=\"exercise\">\r\n<div id=\"fs-id1170572543715\" class=\"textbox\">\r\n<p id=\"fs-id1170572543717\"><strong>58.\u00a0<\/strong>[latex]\\log_4 (x+5)=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572543761\" class=\"exercise\">\r\n<div id=\"fs-id1170572543763\" class=\"textbox\">\r\n<p id=\"fs-id1170572543765\"><strong>59.\u00a0<\/strong>[latex]\\log(2x-7)=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572543794\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572543794\"]\r\n<p id=\"fs-id1170572543794\">[latex]x=4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572543807\" class=\"exercise\">\r\n<div id=\"fs-id1170572543809\" class=\"textbox\">\r\n<p id=\"fs-id1170572543811\"><strong>60.\u00a0<\/strong>[latex]\\ln\\sqrt{x+3}=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572088346\" class=\"exercise\">\r\n<div id=\"fs-id1170572088348\" class=\"textbox\">\r\n<p id=\"fs-id1170572088350\"><strong>61.\u00a0<\/strong>[latex]\\log_6 (x+9)+\\log_6 x=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572088391\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572088391\"]\r\n<p id=\"fs-id1170572088391\">[latex]x=3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572088404\" class=\"exercise\">\r\n<div id=\"fs-id1170572088406\" class=\"textbox\">\r\n<p id=\"fs-id1170572088408\"><strong>62.\u00a0<\/strong>[latex]\\log_4(x+2)-\\log_4(x-1)=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572088466\" class=\"exercise\">\r\n<div id=\"fs-id1170572088468\" class=\"textbox\">\r\n<p id=\"fs-id1170572088470\"><strong>63.\u00a0<\/strong>[latex]\\ln x+\\ln (x-2)=\\ln 4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571120347\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571120347\"]\r\n<p id=\"fs-id1170571120347\">[latex]1+\\sqrt{5}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572088504\">For the following exercises, use the change-of-base formula and either base 10 or base [latex]e[\/latex] to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.<\/p>\r\n\r\n<div id=\"fs-id1170572088513\" class=\"exercise\">\r\n<div id=\"fs-id1170572088515\" class=\"textbox\">\r\n<p id=\"fs-id1170572088517\"><strong>64.\u00a0<\/strong>[latex]\\log_5 47[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572088562\" class=\"exercise\">\r\n<div id=\"fs-id1170572088564\" class=\"textbox\">\r\n<p id=\"fs-id1170572088566\"><strong>65.\u00a0<\/strong>[latex]\\log_7 82[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572088582\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572088582\"]\r\n<p id=\"fs-id1170572088582\">[latex]\\frac{\\ln 82}{\\ln 7} \\approx 2.2646[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572227257\" class=\"exercise\">\r\n<div id=\"fs-id1170572227259\" class=\"textbox\">\r\n<p id=\"fs-id1170572227261\"><strong>66.\u00a0<\/strong>[latex]\\log_6 103[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572227306\" class=\"exercise\">\r\n<div id=\"fs-id1170572227308\" class=\"textbox\">\r\n<p id=\"fs-id1170572227310\"><strong>67.\u00a0<\/strong>[latex]\\log_{0.5} 211[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572227327\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572227327\"]\r\n<p id=\"fs-id1170572227327\">[latex]\\frac{\\ln 211}{\\ln 0.5} \\approx -7.7211[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572227358\" class=\"exercise\">\r\n<div id=\"fs-id1170572227360\" class=\"textbox\">\r\n<p id=\"fs-id1170572227362\"><strong>68.\u00a0<\/strong>[latex]{\\text{log}}_{2}\\pi [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572227407\" class=\"exercise\">\r\n<div id=\"fs-id1170572227409\" class=\"textbox\">\r\n<p id=\"fs-id1170572227411\"><strong>69.\u00a0<\/strong>[latex]\\log_{0.2} 0.452[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572227428\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572227428\"]\r\n<p id=\"fs-id1170572227428\">[latex]\\frac{\\ln 0.452}{\\ln 0.2} \\approx 0.4934[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572227457\" class=\"exercise\">\r\n<div id=\"fs-id1170572227459\" class=\"textbox\">\r\n<p id=\"fs-id1170572227461\"><strong>70.\u00a0<\/strong>Rewrite the following expressions in terms of exponentials and simplify.<\/p>\r\n<p id=\"fs-id1170572227465\">a. [latex]2\\cosh(\\ln x)[\/latex]<\/p>\r\nb. [latex]\\cosh 4x+\\sinh 4x[\/latex]\r\n\r\nc. [latex]\\cosh 2x-\\sinh 2x[\/latex]\r\n\r\nd. [latex]\\ln(\\cosh x+\\sinh x)+\\ln(\\cosh x-\\sinh x)[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572230541\" class=\"exercise\">\r\n<div id=\"fs-id1170572230543\" class=\"textbox\">\r\n<p id=\"fs-id1170572230545\"><strong>71. [T]<\/strong> The number of bacteria [latex]N[\/latex]\u00a0in a culture after [latex]t[\/latex] days can be modeled by the function [latex]N(t)=1300\u00b72^{t\/4}[\/latex]. Find the number of bacteria present after 15 days.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572230605\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572230605\"]\r\n<p id=\"fs-id1170572230605\">[latex]\\approx 17,491[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572230620\" class=\"exercise\">\r\n<div id=\"fs-id1170572230622\" class=\"textbox\">\r\n<p id=\"fs-id1170572230624\"><strong>72. [T]<\/strong> The demand [latex]D[\/latex] (in millions of barrels) for oil in an oil-rich country is given by the function [latex]D(p)=150\u00b72.7^{-0.25p}[\/latex], where [latex]p[\/latex] is the price (in dollars) of a barrel of oil. Find the amount of oil demanded (to the nearest million barrels) when the price is between $15 and $20.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572230690\" class=\"exercise\">\r\n<div id=\"fs-id1170572230692\" class=\"textbox\">\r\n<p id=\"fs-id1170572230695\"><strong>73. [T]<\/strong> The accumulated amount [latex]A[\/latex] of a $100,000 investment whose interest compounds continuously for [latex]t[\/latex] years is given by [latex]A(t)=100,000\u00b7e^{0.055t}[\/latex]. Find the amount [latex]A[\/latex] accumulated in 5 years.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572472846\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572472846\"]\r\n<p id=\"fs-id1170572472846\">Approximately $131,653 is accumulated in 5 years.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572472851\" class=\"exercise\">\r\n<div id=\"fs-id1170572472853\" class=\"textbox\">\r\n<p id=\"fs-id1170572472855\"><strong>74. [T]<\/strong> An investment is compounded monthly, quarterly, or yearly and is given by the function [latex]A=P(1+\\frac{j}{n})^{nt}[\/latex], where [latex]A[\/latex] is the value of the investment at time [latex]t, \\, P[\/latex] is the initial principle that was invested, [latex]j[\/latex] is the annual interest rate, and [latex]n[\/latex] is the number of time the interest is compounded per year. Given a yearly interest rate of 3.5% and an initial principle of $100,000, find the amount [latex]A[\/latex] accumulated in 5 years for interest that is compounded a. daily, b., monthly, c. quarterly, and d. yearly.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572472938\" class=\"exercise\">\r\n<div id=\"fs-id1170572472940\" class=\"textbox\">\r\n<p id=\"fs-id1170572472942\"><strong>75. [T]<\/strong> The concentration of hydrogen ions in a substance is denoted by [latex][\\text{H}^{+}][\/latex], measured in moles per liter. The pH of a substance is defined by the logarithmic function [latex]\\text{pH}=\u2212\\log[\\text{H}^{+}][\/latex]. This function is used to measure the acidity of a substance. The pH of water is 7. A substance with a pH less than 7 is an acid, whereas one that has a pH of more than 7 is a base.<\/p>\r\n\r\n<ol id=\"fs-id1170572472994\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Find the pH of the following substances. Round answers to one digit.<\/li>\r\n \t<li>Determine whether the substance is an acid or a base.\r\n<ol id=\"fs-id1170572473006\">\r\n \t<li>Eggs: [latex][\\text{H}^{+}]=1.6 \\times 10^{-8}[\/latex] mol\/L<\/li>\r\n \t<li>Beer: [latex][\\text{H}^{+}]=3.16 \\times 10^{-3}[\/latex] mol\/L<\/li>\r\n \t<li>Tomato Juice: [latex][\\text{H}^{+}]=7.94 \\times 10^{-5}[\/latex] mol\/L<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572296436\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572296436\"]\r\n<p id=\"fs-id1170572296436\">i. a. pH = 8 b. Base ii. a. pH = 3 b. Acid iii. a. pH = 4 b. Acid<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572296441\" class=\"exercise\">\r\n<div id=\"fs-id1170572296443\" class=\"textbox\">\r\n<p id=\"fs-id1170572296445\"><strong>76. [T]<\/strong> Iodine-131 is a radioactive substance that decays according to the function [latex]Q(t)=Q_0\u00b7e^{-0.08664t}[\/latex], where [latex]Q_0[\/latex] is the initial quantity of a sample of the substance and [latex]t[\/latex] is in days. Determine how long it takes (to the nearest day) for 95% of a quantity to decay.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572296509\" class=\"exercise\">\r\n<div id=\"fs-id1170572296511\" class=\"textbox\">\r\n<p id=\"fs-id1170572296513\"><strong>77. [T]<\/strong> According to the World Bank, at the end of 2013 ([latex]t=0[\/latex]) the U.S. population was 316 million and was increasing according to the following model:<\/p>\r\n<p id=\"fs-id1170572296531\">[latex]P(t)=316e^{0.0074t}[\/latex],<\/p>\r\n<p id=\"fs-id1170572296560\">where [latex]P[\/latex] is measured in millions of people and [latex]t[\/latex] is measured in years after 2013.<\/p>\r\n\r\n<ol id=\"fs-id1170572296574\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Based on this model, what will be the population of the United States in 2020?<\/li>\r\n \t<li>Determine when the U.S. population will be twice what it is in 2013.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572296591\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572296591\"]\r\n<p id=\"fs-id1170572296591\">a. [latex]\\approx 333[\/latex] million b. 94 years from 2013, or in 2107<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572296604\" class=\"exercise\">\r\n<div id=\"fs-id1170572296606\" class=\"textbox\">\r\n<p id=\"fs-id1170572296608\"><strong>78. [T]<\/strong> The amount [latex]A[\/latex] accumulated after 1000 dollars is invested for [latex]t[\/latex] years at an interest rate of 4% is modeled by the function [latex]A(t)=1000(1.04)^t[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1170572296659\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Find the amount accumulated after 5 years and 10 years.<\/li>\r\n \t<li>Determine how long it takes for the original investment to triple.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572296689\" class=\"exercise\">\r\n<div id=\"fs-id1170572296691\" class=\"textbox\">\r\n<p id=\"fs-id1170572296693\"><strong>79. [T]<\/strong> A bacterial colony grown in a lab is known to double in number in 12 hours. Suppose, initially, there are 1000 bacteria present.<\/p>\r\n\r\n<ol id=\"fs-id1170572212877\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Use the exponential function [latex]Q=Q_0e^{kt}[\/latex] to determine the value [latex]k[\/latex], which is the growth rate of the bacteria. Round to four decimal places.<\/li>\r\n \t<li>Determine approximately how long it takes for 200,000 bacteria to grow.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572212922\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572212922\"]\r\n<p id=\"fs-id1170572212922\">a. [latex]k \\approx 0.0578[\/latex] b. [latex]\\approx 92[\/latex] hours<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572212944\" class=\"exercise\">\r\n<div id=\"fs-id1170572212946\" class=\"textbox\">\r\n<p id=\"fs-id1170572212949\"><strong>80. [T]<\/strong> The rabbit population on a game reserve doubles every 6 months. Suppose there were 120 rabbits initially.<\/p>\r\n\r\n<ol id=\"fs-id1170572212957\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Use the exponential function [latex]P=P_0 a^t[\/latex] to determine the growth rate constant [latex]a[\/latex]. Round to four decimal places.<\/li>\r\n \t<li>Use the function in part a. to determine approximately how long it takes for the rabbit population to reach 3500.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572213021\" class=\"exercise\">\r\n<div id=\"fs-id1170572213023\" class=\"textbox\">\r\n<p id=\"fs-id1170572213025\"><strong>81. [T]<\/strong> The 1906 earthquake in San Francisco had a magnitude of 8.3 on the Richter scale. At the same time, in Japan, an earthquake with magnitude 4.9 caused only minor damage. Approximately how much more energy was released by the San Francisco earthquake than by the Japanese earthquake?<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572213038\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572213038\"]\r\n<p id=\"fs-id1170572213038\">The San Francisco earthquake had [latex]10^{3.4}[\/latex] or [latex]\\approx 2512[\/latex] times more energy than the Japan earthquake.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572213065\" class=\"review-exercises\"><\/div>\r\n<div class=\"textbox shaded\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170572544592\" class=\"definition\">\r\n \t<dt>base<\/dt>\r\n \t<dd id=\"fs-id1170572544597\">the number [latex]b[\/latex] in the exponential function [latex]f(x)=b^x[\/latex] and the logarithmic function [latex]f(x)=\\log_b x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n \t<dt>exponent<\/dt>\r\n \t<dd id=\"fs-id1170572544654\">the value [latex]x[\/latex] in the expression [latex]b^x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544670\" class=\"definition\">\r\n \t<dt>hyperbolic functions<\/dt>\r\n \t<dd id=\"fs-id1170572544676\">the functions denoted [latex]\\sinh, \\, \\cosh, \\, \\tanh, \\, \\text{csch}, \\, \\text{sech}[\/latex], and [latex]\\coth[\/latex], which involve certain combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572294452\" class=\"definition\">\r\n \t<dt>inverse hyperbolic functions<\/dt>\r\n \t<dd id=\"fs-id1170572294458\">the inverses of the hyperbolic functions where [latex]\\cosh[\/latex] and [latex]\\text{sech}[\/latex] are restricted to the domain [latex][0,\\infty)[\/latex]; each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572294493\" class=\"definition\">\r\n \t<dt>natural exponential function<\/dt>\r\n \t<dd id=\"fs-id1170572294499\">the function [latex]f(x)=e^x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572294522\" class=\"definition\">\r\n \t<dt>natural logarithm<\/dt>\r\n \t<dd id=\"fs-id1170572294527\">the function [latex]\\ln x=\\log_e x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572294549\" class=\"definition\">\r\n \t<dt>number e<\/dt>\r\n \t<dd id=\"fs-id1170572294554\">as [latex]m[\/latex] gets larger, the quantity [latex](1+(1\/m))^m[\/latex] gets closer to some real number; we define that real number to be [latex]e[\/latex]; the value of [latex]e[\/latex] is approximately 2.718282<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Identify the form of an exponential function.<\/li>\n<li>Explain the difference between the graphs of [latex]x^b[\/latex] and [latex]b^x[\/latex].<\/li>\n<li>Recognize the significance of the number [latex]e[\/latex].<\/li>\n<li>Identify the form of a logarithmic function.<\/li>\n<li>Explain the relationship between exponential and logarithmic functions.<\/li>\n<li>Describe how to calculate a logarithm to a different base.<\/li>\n<li>Identify the hyperbolic functions, their graphs, and basic identities.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1170572094366\">In this section we examine exponential and logarithmic functions. We use the properties of these functions to solve equations involving exponential or logarithmic terms, and we study the meaning and importance of the number [latex]e[\/latex]. We also define hyperbolic and inverse hyperbolic functions, which involve combinations of exponential and logarithmic functions. (Note that we present alternative definitions of exponential and logarithmic functions in the chapter <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/introduction-6\/\">Applications of Integrations<\/a>, and prove that the functions have the same properties with either definition.)<\/p>\n<div id=\"fs-id1170572108759\" class=\"bc-section section\">\n<h1>Exponential Functions<\/h1>\n<p id=\"fs-id1170572246259\">Exponential functions arise in many applications. One common example is <span class=\"no-emphasis\">population growth<\/span>.<\/p>\n<p id=\"fs-id1170572449480\">For example, if a population starts with [latex]P_0[\/latex] individuals and then grows at an annual rate of [latex]2\\%[\/latex], its population after 1 year is<\/p>\n<div class=\"equation unnumbered\">[latex]P(1)=P_0+0.02P_0=P_0(1+0.02)=P_0(1.02)[\/latex].<\/div>\n<p id=\"fs-id1170572092410\">Its population after 2 years is<\/p>\n<div id=\"fs-id1170572177937\" class=\"equation unnumbered\">[latex]P(2)=P(1)+0.02P(1)=P(1)(1.02)=P_0(1.02)^2[\/latex].<\/div>\n<p id=\"fs-id1170572130048\">In general, its population after [latex]t[\/latex] years is<\/p>\n<div id=\"fs-id1170572280288\" class=\"equation unnumbered\">[latex]P(t)=P_0(1.02)^t[\/latex],<\/div>\n<p>which is an exponential function. More generally, any function of the form [latex]f(x)=b^x[\/latex], where [latex]b>0, \\, b \\ne 1[\/latex], is an exponential function with <strong>base<\/strong> [latex]b[\/latex] and <strong>exponent<\/strong> [latex]x[\/latex]. Exponential functions have constant bases and variable exponents. Note that a function of the form [latex]f(x)=x^b[\/latex] for some constant [latex]b[\/latex] is not an exponential function but a power function.<\/p>\n<p id=\"fs-id1170572248051\">To see the difference between an exponential function and a power function, we compare the functions [latex]y=x^2[\/latex] and [latex]y=2^x[\/latex]. In <a class=\"autogenerated-content\" href=\"#fs-id1170572205233\">(Figure)<\/a>, we see that both [latex]2^x[\/latex] and [latex]x^2[\/latex] approach infinity as [latex]x \\to \\infty[\/latex]. Eventually, however, [latex]2^x[\/latex] becomes larger than [latex]x^2[\/latex] and grows more rapidly as [latex]x \\to \\infty[\/latex]. In the opposite direction, as [latex]x \\to \u2212\\infty, \\, x^2 \\to \\infty[\/latex], whereas [latex]2^x \\to 0[\/latex]. The line [latex]y=0[\/latex] is a horizontal asymptote for [latex]y=2^x[\/latex].<\/p>\n<table id=\"fs-id1170572205233\" class=\"column-header\" style=\"width: 779px\" summary=\"A table with 3 rows and 10 columns. The first row is labeled \u201cx\u201d and has the values \u201c-3; -2; -1; 0; 1; 2; 3; 4; 5; 6\u201d. The second row is labeled \u201cx squared\u201d and has the values \u201c9; 4; 1; 0; 1; 4; 9; 16; 25; 36\u201d. The third row is labeled \u201c2 to the power of x\u201d and has the values \u201c(1\/8); (1\/4); (1\/2); 1; 2; 4; 8; 16; 32; 64\u201d.\">\n<caption>Values of [latex]x^2[\/latex] and [latex]2^x[\/latex]<\/caption>\n<tbody>\n<tr valign=\"top\">\n<td style=\"width: 157px\">[latex]\\mathbf{x}[\/latex]<\/td>\n<td style=\"width: 138px\">-3<\/td>\n<td style=\"width: 138px\">-2<\/td>\n<td style=\"width: 138px\">-1<\/td>\n<td style=\"width: 9px\">0<\/td>\n<td style=\"width: 9px\">1<\/td>\n<td style=\"width: 9px\">2<\/td>\n<td style=\"width: 9px\">3<\/td>\n<td style=\"width: 17px\">4<\/td>\n<td style=\"width: 17px\">5<\/td>\n<td style=\"width: 17px\">6<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 157px\">[latex]\\mathbf{x^2}[\/latex]<\/td>\n<td style=\"width: 138px\">9<\/td>\n<td style=\"width: 138px\">4<\/td>\n<td style=\"width: 138px\">1<\/td>\n<td style=\"width: 9px\">0<\/td>\n<td style=\"width: 9px\">1<\/td>\n<td style=\"width: 9px\">4<\/td>\n<td style=\"width: 9px\">9<\/td>\n<td style=\"width: 17px\">16<\/td>\n<td style=\"width: 17px\">25<\/td>\n<td style=\"width: 17px\">36<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 157px\">[latex]\\mathbf{2^x}[\/latex]<\/td>\n<td style=\"width: 138px\">[latex]1\/8[\/latex]<\/td>\n<td style=\"width: 138px\">[latex]1\/4[\/latex]<\/td>\n<td style=\"width: 138px\">[latex]1\/2[\/latex]<\/td>\n<td style=\"width: 9px\">1<\/td>\n<td style=\"width: 9px\">2<\/td>\n<td style=\"width: 9px\">4<\/td>\n<td style=\"width: 9px\">8<\/td>\n<td style=\"width: 17px\">16<\/td>\n<td style=\"width: 17px\">32<\/td>\n<td style=\"width: 17px\">64<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170572247756\">In <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_001\">(Figure)<\/a>, we graph both [latex]y=x^2[\/latex] and [latex]y=2^x[\/latex] to show how the graphs differ.<\/p>\n<div id=\"CNX_Calc_Figure_01_05_001\" class=\"wp-caption aligncenter\">\n<div style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202644\/CNX_Calc_Figure_01_05_001.jpg\" alt=\"An image of a graph. The x axis runs from -10 to 10 and the y axis runs from 0 to 50. The graph is of two functions. The first function is \u201cy = x squared\u201d, which is a parabola. The function decreases until it hits the origin and then begins increasing. The second function is \u201cy = 2 to the power of x\u201d, which starts slightly above the x axis, and begins increasing very rapidly, more rapidly than the first function.\" width=\"325\" height=\"427\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 1.<\/strong> Both [latex]2^x[\/latex] and [latex]x^2[\/latex] approach infinity as [latex]x \\to \\infty[\/latex], but [latex]2^x[\/latex] grows more rapidly than [latex]x^2[\/latex]. As [latex]x \\to \u2212\\infty, \\, x^2 \\to \\infty[\/latex], whereas [latex]2^x \\to 0[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572135350\" class=\"bc-section section\">\n<h2>Evaluating Exponential Functions<\/h2>\n<p id=\"fs-id1170572134608\">Recall the properties of exponents: If [latex]x[\/latex] is a positive integer, then we define [latex]b^x=b\u00b7b \\cdots b[\/latex] (with [latex]x[\/latex] factors of [latex]b[\/latex]). If [latex]x[\/latex] is a negative integer, then [latex]x=\u2212y[\/latex] for some positive integer [latex]y[\/latex], and we define [latex]b^x=b^{\u2212y}=1\/b^y[\/latex]. Also, [latex]b^0[\/latex] is defined to be 1. If [latex]x[\/latex] is a rational number, then [latex]x=p\/q[\/latex], where [latex]p[\/latex] and [latex]q[\/latex] are integers and [latex]b^x=b^{p\/q}=\\sqrt[q]{b^p}[\/latex]. For example, [latex]9^{3\/2}=\\sqrt{9^3}=27[\/latex]. However, how is [latex]b^x[\/latex] defined if [latex]x[\/latex] is an irrational number? For example, what do we mean by [latex]2^{\\sqrt{2}}[\/latex]? This is too complex a question for us to answer fully right now; however, we can make an approximation. In <a class=\"autogenerated-content\" href=\"#fs-id1170572480690\">(Figure)<\/a>, we list some rational numbers approaching [latex]\\sqrt{2}[\/latex], and the values of [latex]2^x[\/latex] for each rational number [latex]x[\/latex] are presented as well. We claim that if we choose rational numbers [latex]x[\/latex] getting closer and closer to [latex]\\sqrt{2}[\/latex], the values of [latex]2^x[\/latex] get closer and closer to some number [latex]L[\/latex]. We define that number [latex]L[\/latex] to be [latex]2^{\\sqrt{2}}[\/latex].<\/p>\n<table id=\"fs-id1170572480690\" class=\"column-header\" summary=\"A table with 2 rows and 6 columns. The first row is labeled \u201cx\u201d and has the values \u201c1.4; 1.41; 1.414; 1.4142; 1.41421; 1.414213\u201d. The second row is labeled \u201c2 to the power of x\u201d and has the values \u201c2.639; 2.65737; 2.66475; 2.665119; 2.665138; 2.665143\u201d.\">\n<caption>Values of [latex]2^x[\/latex] for a List of Rational Numbers Approximating [latex]\\sqrt{2}[\/latex]<\/caption>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]\\mathbf{x}[\/latex]<\/td>\n<td>1.4<\/td>\n<td>1.41<\/td>\n<td>1.414<\/td>\n<td>1.4142<\/td>\n<td>1.41421<\/td>\n<td>1.414213<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\mathbf{2^x}[\/latex]<\/td>\n<td>2.639<\/td>\n<td>2.65737<\/td>\n<td>2.66475<\/td>\n<td>2.665119<\/td>\n<td>2.665138<\/td>\n<td>2.665143<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1170572169644\" class=\"textbox examples\">\n<h3>Bacterial Growth<\/h3>\n<div id=\"fs-id1170572169649\" class=\"exercise\">\n<div id=\"fs-id1170572169651\" class=\"textbox\">\n<p id=\"fs-id1170572169653\">Suppose a particular population of bacteria is known to double in size every 4 hours. If a culture starts with 1000 bacteria, the number of bacteria after 4 hours is [latex]n(4)=1000\u00b72[\/latex]. The number of bacteria after 8 hours is [latex]n(8)=n(4)\u00b72=1000\u00b72^2[\/latex]. In general, the number of bacteria after [latex]4m[\/latex] hours is [latex]n(4m)=1000\u00b72^m[\/latex]. Letting [latex]t=4m[\/latex], we see that the number of bacteria after [latex]t[\/latex] hours is [latex]n(t)=1000\u00b72^{t\/4}[\/latex]. Find the number of bacteria after 6 hours, 10 hours, and 24 hours.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572550969\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572550969\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572550969\">The number of bacteria after 6 hours is given by [latex]n(6)=1000\u00b72^{6\/4} \\approx 2828[\/latex] bacteria. The number of bacteria after 10 hours is given by [latex]n(10)=1000\u00b72^{10\/4} \\approx 5657[\/latex] bacteria. The number of bacteria after 24 hours is given by [latex]n(24)=1000\u00b72^6=64,000[\/latex] bacteria.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572173708\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572173711\" class=\"exercise\">\n<div id=\"fs-id1170572173713\" class=\"textbox\">\n<p id=\"fs-id1170572173715\">Given the exponential function [latex]f(x)=100\u00b73^{x\/2}[\/latex], evaluate [latex]f(4)[\/latex] and [latex]f(10)[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572173781\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572173781\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572173781\">[latex]f(4)=900; \\, f(10)=24,300[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572173826\" class=\"textbox tryit media-2\">\n<p id=\"fs-id1170572173829\">Go to <a href=\"http:\/\/www.openstaxcollege.org\/l\/20_exponengrow\">World Population Balance<\/a> for another example of exponential population growth.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572554025\" class=\"bc-section section\">\n<h2>Graphing Exponential Functions<\/h2>\n<p id=\"fs-id1170572554031\">For any base [latex]b>0, \\, b\\ne 1[\/latex], the exponential function [latex]f(x)=b^x[\/latex] is defined for all real numbers [latex]x[\/latex] and [latex]b^x>0[\/latex]. Therefore, the domain of [latex]f(x)=b^x[\/latex] is [latex](\u2212\\infty ,\\infty)[\/latex] and the range is [latex](0,\\infty)[\/latex]. To graph [latex]b^x[\/latex], we note that for [latex]b>1, \\, b^x[\/latex] is increasing on [latex](\u2212\\infty ,\\infty)[\/latex] and [latex]b^x \\to \\infty[\/latex] as [latex]x \\to \\infty[\/latex], whereas [latex]b^x \\to 0[\/latex] as [latex]x \\to \u2212\\infty[\/latex]. On the other hand, if [latex]0<b<1, \\, f(x)=b^x[\/latex] is decreasing on [latex](\u2212\\infty ,\\infty)[\/latex] and [latex]b^x \\to 0[\/latex] as [latex]x \\to \\infty[\/latex] whereas [latex]b^x \\to \\infty[\/latex] as [latex]x \\to \u2212\\infty[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_002\">(Figure)<\/a>).<\/p>\n<div id=\"CNX_Calc_Figure_01_05_002\" class=\"wp-caption aligncenter\">\n<div style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202647\/CNX_Calc_Figure_01_05_002.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 3 and the y axis runs from 0 to 4. The graph is of four functions. The first function is \u201cf(x) = 2 to the power of x\u201d, an increasing curved function, which starts slightly above the x axis and begins increasing. The second function is \u201cf(x) = 4 to the power of x\u201d, an increasing curved function, which starts slightly above the x axis and begins increasing rapidly, more rapidly than the first function. The third function is \u201cf(x) = (1\/2) to the power of x\u201d, a decreasing curved function with decreases until it gets close to the x axis without touching it. The third function is \u201cf(x) = (1\/4) to the power of x\u201d, a decreasing curved function with decreases until it gets close to the x axis without touching it. It decrases at a faster rate than the third function.\" width=\"325\" height=\"221\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 2.<\/strong> If [latex]b&gt;1[\/latex], then [latex]b^x[\/latex] is increasing on [latex](\u2212\\infty ,\\infty)[\/latex]. If [latex]0&lt;b&lt;1[\/latex], then [latex]b^x[\/latex] is decreasing on [latex](\u2212\\infty ,\\infty)[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572481215\" class=\"textbox tryit media-2\">\n<p id=\"fs-id1170572481218\">Visit this <a href=\"http:\/\/www.openstaxcollege.org\/l\/20_inverse\">site<\/a> for more exploration of the graphs of exponential functions.<\/p>\n<\/div>\n<p id=\"fs-id1170572481226\">Note that exponential functions satisfy the general laws of exponents. To remind you of these laws, we state them as rules.<\/p>\n<div id=\"fs-id1170572481231\" class=\"textbox key-takeaways\">\n<h3>Rule: Laws of Exponents<\/h3>\n<p id=\"fs-id1170572481236\">For any constants [latex]a>0, \\, b>0[\/latex], and for all [latex]x[\/latex] and [latex]y[\/latex],<\/p>\n<ol id=\"fs-id1170572481268\">\n<li>[latex]b^x\u00b7b^y=b^{x+y}[\/latex]<\/li>\n<li>[latex]\\large\\frac{b^x}{b^y} \\normalsize = b^{x-y}[\/latex]<\/li>\n<li>[latex](b^x)^y=b^{xy}[\/latex]<\/li>\n<li>[latex](ab)^x=a^x b^x[\/latex]<\/li>\n<li>[latex]\\large \\frac{a^x}{b^x} \\normalsize =(\\frac{a}{b})^x[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170572440093\" class=\"textbox examples\">\n<h3>Using the Laws of Exponents<\/h3>\n<div id=\"fs-id1170572440098\" class=\"exercise\">\n<div id=\"fs-id1170572440100\" class=\"textbox\">\n<p id=\"fs-id1170572440102\">Use the laws of exponents to simplify each of the following expressions.<\/p>\n<ol id=\"fs-id1170572440106\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\large \\frac{(2x^{2\/3})^3}{(4x^{-1\/3})^2}[\/latex]<\/li>\n<li>[latex]\\large \\frac{(x^3 y^{-1})^2}{(xy^2)^{-2}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572453127\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572453127\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572453127\" style=\"list-style-type: lower-alpha\">\n<li>We can simplify as follows:\n<div id=\"fs-id1170570966957\" class=\"equation unnumbered\">[latex]\\large \\frac{(2x^{2\/3})^3}{(4x^{-1\/3})^2} \\normalsize = \\large \\frac{2^3(x^{2\/3})^3}{4^2(x^{-1\/3})^2} \\normalsize = \\large \\frac{8x^2}{16x^{-2\/3}} \\normalsize = \\large \\frac{x^2x^{2\/3}}{2} \\normalsize = \\large \\frac{x^{8\/3}}{2}[\/latex].<\/div>\n<\/li>\n<li>We can simplify as follows:\n<div id=\"fs-id1170573582280\" class=\"equation unnumbered\">[latex]\\large \\frac{(x^3y^{-1})^2}{(xy^2)^{-2}} \\normalsize = \\large \\frac{(x^3)^2(y^{-1})^2}{x^{-2}(y^2)^{-2}} \\normalsize = \\large \\frac{x^6y^{-2}}{x^{-2}y^{-4}} \\normalsize = x^6x^2y^{-2}y^4 = x^8y^2[\/latex].<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises checkpoint\">\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p id=\"fs-id1170572452234\">Use the laws of exponents to simplify [latex](6x^{-3}y^2)\/(12x^{-4}y^5)[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1170572452533\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572452533\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572452533\" class=\"hidden-answer\" style=\"display: none\">[latex]x\/(2y^3)[\/latex]<\/div>\n<div id=\"fs-id1165042836022\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042707513\">[latex]x^a\/x^b=x^{a-b}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572452563\" class=\"bc-section section\">\n<h1>The <strong>Number<\/strong><em><strong>\u00a0[latex]e[\/latex]<\/strong><\/em><\/h1>\n<p id=\"fs-id1170572452572\">A special type of exponential function appears frequently in real-world applications. To describe it, consider the following example of exponential growth, which arises from <span class=\"no-emphasis\">compounding interest<\/span> in a savings account. Suppose a person invests [latex]P[\/latex] dollars in a savings account with an annual interest rate [latex]r[\/latex], compounded annually. The amount of money after 1 year is<\/p>\n<div id=\"fs-id1170572452592\" class=\"equation unnumbered\">[latex]A(1)=P+rP=P(1+r)[\/latex].<\/div>\n<p id=\"fs-id1170572449156\">The amount of money after 2 years is<\/p>\n<div id=\"fs-id1170572449164\" class=\"equation unnumbered\">[latex]A(2)=A(1)+rA(1)=P(1+r)+rP(1+r)=P(1+r)^2[\/latex].<\/div>\n<p id=\"fs-id1170572449268\">More generally, the amount after [latex]t[\/latex] years is<\/p>\n<div id=\"fs-id1170572449276\" class=\"equation unnumbered\">[latex]A(t)=P(1+r)^t[\/latex].<\/div>\n<p id=\"fs-id1170572449316\">If the money is compounded 2 times per year, the amount of money after half a year is<\/p>\n<div id=\"fs-id1170572449320\" class=\"equation unnumbered\">[latex]A(\\frac{1}{2})=P+(\\frac{r}{2})P=P(1+(\\frac{r}{2}))[\/latex].<\/div>\n<p id=\"fs-id1170572548410\">The amount of money after 1 year is<\/p>\n<div id=\"fs-id1170572548417\" class=\"equation unnumbered\">[latex]A(1)=A(\\frac{1}{2})+(\\frac{r}{2})A(\\frac{1}{2})=P(1+\\frac{r}{2})+\\frac{r}{2}(P(1+\\frac{r}{2}))=P(1+\\frac{r}{2})^2[\/latex].<\/div>\n<p id=\"fs-id1170572229281\">After [latex]t[\/latex] years, the amount of money in the account is<\/p>\n<div id=\"fs-id1170572229288\" class=\"equation unnumbered\">[latex]A(t)=P(1+\\frac{r}{2})^{2t}[\/latex].<\/div>\n<p id=\"fs-id1170572229335\">More generally, if the money is compounded [latex]n[\/latex] times per year, the amount of money in the account after [latex]t[\/latex] years is given by the function<\/p>\n<div class=\"equation unnumbered\">[latex]A(t)=P(1+\\frac{r}{n})^{nt}[\/latex].<\/div>\n<p id=\"fs-id1170572229394\">What happens as [latex]n\\to \\infty[\/latex]? To answer this question, we let [latex]m=n\/r[\/latex] and write<\/p>\n<div id=\"fs-id1170572451284\" class=\"equation unnumbered\">[latex](1+\\frac{r}{n})^{nt}=(1+\\frac{1}{m})^{mrt}[\/latex],<\/div>\n<p id=\"fs-id1170572451344\">and examine the behavior of [latex](1+1\/m)^m[\/latex] as [latex]m\\to \\infty[\/latex], using a table of values (<a class=\"autogenerated-content\" href=\"#fs-id1170572451390\">(Figure)<\/a>).<\/p>\n<table id=\"fs-id1170572451390\" class=\"column-header\" summary=\"A table with 2 rows and 6 columns. The first row is labeled \u201cm\u201d and has the values \u201c10; 100; 1000; 10,000; 100,000; 1,000,000\u201d. The second row is labeled \u201c(1 + (1\/m)) to the power of m\u201d and has the values \u201c2.5937; 2.7048; 2.71692; 2.71815; 2.718268; 2.718280\u201d.\">\n<caption>Values of [latex](1+\\frac{1}{m})^m[\/latex] as [latex]m \\to \\infty[\/latex]<\/caption>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]\\mathbf{m}[\/latex]<\/td>\n<td>10<\/td>\n<td>100<\/td>\n<td>1000<\/td>\n<td>10,000<\/td>\n<td>100,000<\/td>\n<td>1,000,000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\mathbf{(1+\\frac{1}{m})^m}[\/latex]<\/td>\n<td>2.5937<\/td>\n<td>2.7048<\/td>\n<td>2.71692<\/td>\n<td>2.71815<\/td>\n<td>2.718268<\/td>\n<td>2.718280<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Looking at this table, it appears that [latex](1+1\/m)^m[\/latex] is approaching a number between 2.7 and 2.8 as [latex]m\\to \\infty[\/latex]. In fact, [latex](1+1\/m)^m[\/latex] does approach some number as [latex]m\\to \\infty[\/latex]. We call this number [latex]e[\/latex]. To six decimal places of accuracy,<\/p>\n<div id=\"fs-id1170572549050\" class=\"equation unnumbered\">[latex]e \\approx 2.718282[\/latex].<\/div>\n<p id=\"fs-id1170572549064\">The letter [latex]e[\/latex] was first used to represent this number by the Swiss mathematician Leonhard Euler during the 1720s. Although Euler did not discover the number, he showed many important connections between [latex]e[\/latex] and logarithmic functions. We still use the notation [latex]e[\/latex] today to honor Euler\u2019s work because it appears in many areas of mathematics and because we can use it in many practical applications.<\/p>\n<p id=\"fs-id1170572549084\">Returning to our savings account example, we can conclude that if a person puts [latex]P[\/latex] dollars in an account at an annual interest rate [latex]r[\/latex], compounded continuously, then [latex]A(t)=Pe^{rt}[\/latex]. This function may be familiar. Since functions involving base [latex]e[\/latex] arise often in applications, we call the function [latex]f(x)=e^x[\/latex] the <strong>natural exponential function<\/strong>. Not only is this function interesting because of the definition of the number [latex]e[\/latex], but also, as discussed next, its graph has an important property.<\/p>\n<p id=\"fs-id1170572451705\">Since [latex]e>1[\/latex], we know [latex]e^x[\/latex] is increasing on [latex](\u2212\\infty ,\\infty)[\/latex]. In <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_003\">(Figure)<\/a>, we show a graph of [latex]f(x)=e^x[\/latex] along with a <em>tangent line<\/em> to the graph of at [latex]x=0[\/latex]. We give a precise definition of tangent line in the next chapter; but, informally, we say a tangent line to a graph of [latex]f[\/latex] at [latex]x=a[\/latex] is a line that passes through the point [latex](a,f(a))[\/latex] and has the same \u201cslope\u201d as [latex]f[\/latex] at that point. The function [latex]f(x)=e^x[\/latex] is the only exponential function [latex]b^x[\/latex] with tangent line at [latex]x=0[\/latex] that has a slope of 1. As we see later in the text, having this property makes the natural exponential function the simplest exponential function to use in many instances.<\/p>\n<div id=\"CNX_Calc_Figure_01_05_003\" class=\"wp-caption aligncenter\">\n<div style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202649\/CNX_Calc_Figure_01_05_003.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 3 and the y axis runs from 0 to 4. The graph is of the function \u201cf(x) = e to power of x\u201d, an increasing curved function that starts slightly above the x axis. The y intercept is at the point (0, 1). At this point, a line is drawn tangent to the function. This line has the label \u201cslope = 1\u201d.\" width=\"325\" height=\"202\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 3.<\/strong> The graph of [latex]f(x)=e^x[\/latex] has a tangent line with slope 1 at [latex]x=0[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572547818\" class=\"textbox examples\">\n<h3>Compounding Interest<\/h3>\n<div id=\"fs-id1170572547823\" class=\"exercise\">\n<div id=\"fs-id1170572547825\" class=\"textbox\">\n<p id=\"fs-id1170572547827\">Suppose [latex]\\$500[\/latex] is invested in an account at an annual interest rate of [latex]r=5.5\\%[\/latex], compounded continuously.<\/p>\n<ol id=\"fs-id1170572547852\" style=\"list-style-type: lower-alpha\">\n<li>Let [latex]t[\/latex] denote the number of years after the initial investment and [latex]A(t)[\/latex] denote the amount of money in the account at time [latex]t[\/latex]. Find a formula for [latex]A(t)[\/latex].<\/li>\n<li>Find the amount of money in the account after 10 years and after 20 years.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572542850\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572542850\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572542850\" style=\"list-style-type: lower-alpha\">\n<li>If [latex]P[\/latex] dollars are invested in an account at an annual interest rate [latex]r[\/latex], compounded continuously, then [latex]A(t)=Pe^{rt}[\/latex]. Here [latex]P=\\$500[\/latex] and [latex]r=0.055[\/latex]. Therefore, [latex]A(t)=500e^{0.055t}[\/latex].<\/li>\n<li>After 10 years, the amount of money in the account is\n<div id=\"fs-id1170573386623\" class=\"equation unnumbered\">[latex]A(10)=500e^{0.055\u00b710}=500e^{0.55}\\approx \\$866.63[\/latex].<\/div>\n<p>After 20 years, the amount of money in the account is<\/p>\n<div id=\"fs-id1170573390418\" class=\"equation unnumbered\">[latex]A(20)=500e^{0.055\u00b720}=500e^{1.1}\\approx \\$1,502.08[\/latex].<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572455119\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572455122\" class=\"exercise\">\n<div id=\"fs-id1170572455124\" class=\"textbox\">\n<p id=\"fs-id1170572455126\">If [latex]\\$750[\/latex] is invested in an account at an annual interest rate of [latex]4\\%[\/latex], compounded continuously, find a formula for the amount of money in the account after [latex]t[\/latex] years. Find the amount of money after 30 years.<\/p>\n<\/div>\n<div class=\"solution\">\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<div class=\"solution\">\n<p id=\"fs-id1170572455190\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q505690\">Show Answer<\/span><\/p>\n<div id=\"q505690\" class=\"hidden-answer\" style=\"display: none\">[latex]A(t)=750e^{0.04t}[\/latex]. After 30 years, there will be approximately [latex]\\$2,490.09[\/latex].<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165039562082\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042707350\">[latex]A(t)=Pe^{rt}[\/latex].<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572455243\" class=\"bc-section section\">\n<h1>Logarithmic Functions<\/h1>\n<p id=\"fs-id1170572455248\">Using our understanding of exponential functions, we can discuss their inverses, which are the logarithmic functions. These come in handy when we need to consider any phenomenon that varies over a wide range of values, such as pH in chemistry or decibels in sound levels.<\/p>\n<p id=\"fs-id1170572455254\">The exponential function [latex]f(x)=b^x[\/latex] is one-to-one, with domain [latex](\u2212\\infty ,\\infty)[\/latex] and range [latex](0,\\infty )[\/latex]. Therefore, it has an inverse function, called the <em>logarithmic function with base<\/em> [latex]b[\/latex]. For any [latex]b>0, \\, b \\ne 1[\/latex], the logarithmic function with base [latex]b[\/latex], denoted [latex]\\log_b[\/latex], has domain [latex](0,\\infty )[\/latex] and range [latex](\u2212\\infty ,\\infty )[\/latex], and satisfies<\/p>\n<div id=\"fs-id1170572551548\" class=\"equation unnumbered\">[latex]\\log_b(x)=y[\/latex] if and only if [latex]b^y=x[\/latex].<\/div>\n<p id=\"fs-id1170572545100\">For example,<\/p>\n<div id=\"fs-id1170572545103\" class=\"equation unnumbered\">[latex]\\begin{array}{cccc} \\log_2 (8)=3\\hfill & & & \\text{since}\\phantom{\\rule{3em}{0ex}}2^3=8,\\hfill \\\\ \\log_{10} (\\frac{1}{100})=-2\\hfill & & & \\text{since}\\phantom{\\rule{3em}{0ex}}10^{-2}=\\frac{1}{10^2}=\\frac{1}{100},\\hfill \\\\ \\log_b (1)=0\\hfill & & & \\text{since}\\phantom{\\rule{3em}{0ex}}b^0=1 \\, \\text{for any base} \\, b>0.\\hfill \\end{array}[\/latex]<\/div>\n<p>Furthermore, since [latex]y=\\log_b (x)[\/latex] and [latex]y=b^x[\/latex] are inverse functions,<\/p>\n<div id=\"fs-id1170572169121\" class=\"equation unnumbered\">[latex]\\log_b (b^x)=x \\, \\text{and} \\, b^{\\log_b (x)}=x[\/latex].<\/div>\n<p id=\"fs-id1170572169184\">The most commonly used logarithmic function is the function [latex]\\log_e (x)[\/latex]. Since this function uses natural [latex]e[\/latex] as its base, it is called the<strong> natural logarithm<\/strong>. Here we use the notation [latex]\\ln(x)[\/latex] or [latex]\\ln x[\/latex] to mean [latex]\\log_e (x)[\/latex]. For example,<\/p>\n<div id=\"fs-id1170572243723\" class=\"equation unnumbered\">[latex]\\ln (e)=\\log_e (e)=1, \\, \\ln(e^3)=\\log_e (e^3)=3, \\, \\ln(1)=\\log_e (1)=0[\/latex].<\/div>\n<p id=\"fs-id1170572243836\">Since the functions [latex]f(x)=e^x[\/latex] and [latex]g(x)=\\ln(x)[\/latex] are inverses of each other,<\/p>\n<div id=\"fs-id1170572243887\" class=\"equation unnumbered\">[latex]\\ln(e^x)=x \\, \\text{and} \\, e^{\\ln x}=x[\/latex],<\/div>\n<p id=\"fs-id1170572229068\">and their graphs are symmetric about the line [latex]y=x[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_004\">(Figure)<\/a>).<\/p>\n<div id=\"CNX_Calc_Figure_01_05_004\" class=\"wp-caption aligncenter\">\n<div style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202652\/CNX_Calc_Figure_01_05_004.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -3 to 4. The graph is of two functions. The first function is \u201cf(x) = e to power of x\u201d, an increasing curved function that starts slightly above the x axis. The y intercept is at the point (0, 1) and there is no x intercept. The second function is \u201cf(x) = ln(x)\u201d, an increasing curved function. The x intercept is at the point (1, 0) and there is no y intercept. A dotted line with label \u201cy = x\u201d is also plotted on the graph, to show that the functions are mirror images over this line.\" width=\"325\" height=\"312\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4: The functions [latex]y=e^x[\/latex] and [latex]y=\\ln(x)[\/latex] are inverses of each other, so their graphs are symmetric about the line [latex]y=x[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572229152\" class=\"textbox tryit media-2\">\n<p id=\"fs-id1170572229156\">At this <a href=\"http:\/\/www.openstaxcollege.org\/l\/20_logscale\">site<\/a> you can see an example of a base-10 logarithmic scale.<\/p>\n<\/div>\n<p id=\"fs-id1170572229166\">In general, for any base [latex]b>0, \\, b\\ne 1[\/latex], the function [latex]g(x)=\\log_b (x)[\/latex] is symmetric about the line [latex]y=x[\/latex] with the function [latex]f(x)=b^x[\/latex]. Using this fact and the graphs of the exponential functions, we graph functions [latex]\\log_b (x)[\/latex] for several values of [latex]b>1[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_005\">(Figure)<\/a>).<\/p>\n<div id=\"CNX_Calc_Figure_01_05_005\" class=\"wp-caption aligncenter\">\n<div style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202654\/CNX_Calc_Figure_01_05_005.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 3 and the y axis runs from 0 to 4. The graph is of three functions. All three functions a log functions that are increasing curved functions that start slightly to the right of the y axis and have an x intercept at (1, 0). The first function is \u201cy = log base 10 (x)\u201d, the second function is \u201cf(x) = ln(x)\u201d, and the third function is \u201cy = log base 2 (x)\u201d. The third function increases the most rapidly, the second function increases next most rapidly, and the third function increases the slowest.\" width=\"325\" height=\"312\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 5: Graphs of [latex]y=\\log_b (x)[\/latex] are depicted for [latex]b=2, \\, e, \\, 10[\/latex].<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572482697\">Before solving some equations involving exponential and logarithmic functions, let\u2019s review the basic properties of logarithms.<\/p>\n<div id=\"fs-id1170572482702\" class=\"textbox key-takeaways\">\n<h3>Rule: Properties of Logarithms<\/h3>\n<p id=\"fs-id1170572482707\">If [latex]a,b,c>0, \\, b\\ne 1[\/latex], and [latex]r[\/latex] is any real number, then<\/p>\n<div id=\"fs-id1170572482744\" class=\"equation unnumbered\">[latex]\\begin{array}{cccc}1.\\phantom{\\rule{2em}{0ex}}\\log_b (ac)=\\log_b (a)+\\log_b (c)\\hfill & & & \\text{(Product property)}\\hfill \\\\ 2.\\phantom{\\rule{2em}{0ex}}\\log_b(\\frac{a}{c})=\\log_b (a) -\\log_b (c)\\hfill & & & \\text{(Quotient property)}\\hfill \\\\ 3.\\phantom{\\rule{2em}{0ex}}\\log_b (a^r)=r \\log_b (a)\\hfill & & & \\text{(Power property)}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1170572551970\" class=\"textbox examples\">\n<h3>Solving Equations Involving Exponential Functions<\/h3>\n<div id=\"fs-id1170572551976\" class=\"exercise\">\n<div id=\"fs-id1170572551978\" class=\"textbox\">\n<p id=\"fs-id1170572551980\">Solve each of the following equations for [latex]x[\/latex].<\/p>\n<ol id=\"fs-id1170572551988\" style=\"list-style-type: lower-alpha\">\n<li>[latex]5^x=2[\/latex]<\/li>\n<li>[latex]e^x+6e^{\u2212x}=5[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572550555\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572550555\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572550555\" style=\"list-style-type: lower-alpha\">\n<li>Applying the natural logarithm function to both sides of the equation, we have\n<div id=\"fs-id1170571071223\" class=\"equation unnumbered\">[latex]\\ln 5^x=\\ln 2[\/latex].<\/div>\n<p>Using the power property of logarithms,<\/p>\n<div id=\"fs-id1170571277779\" class=\"equation unnumbered\">[latex]x \\ln 5=\\ln 2[\/latex].<\/div>\n<p>Therefore, [latex]x=\\ln 2 \/ \\ln 5[\/latex].<\/li>\n<li>Multiplying both sides of the equation by [latex]e^x[\/latex], we arrive at the equation\n<div id=\"fs-id1170571301573\" class=\"equation unnumbered\">[latex]e^{2x}+6=5e^x[\/latex].<\/div>\n<p>Rewriting this equation as<\/p>\n<div id=\"fs-id1170573367583\" class=\"equation unnumbered\">[latex]e^{2x}-5e^x+6=0[\/latex],<\/div>\n<p>we can then rewrite it as a quadratic equation in [latex]e^x[\/latex]:<\/p>\n<div id=\"fs-id1170570976384\" class=\"equation unnumbered\">[latex](e^x)^2-5(e^x)+6=0[\/latex].<\/div>\n<p>Now we can solve the quadratic equation. Factoring this equation, we obtain<\/p>\n<div id=\"fs-id1170573400246\" class=\"equation unnumbered\">[latex](e^x-3)(e^x-2)=0[\/latex].<\/div>\n<p>Therefore, the solutions satisfy [latex]e^x=3[\/latex] and [latex]e^x=2[\/latex]. Taking the natural logarithm of both sides gives us the solutions [latex]x=\\ln 3, \\, \\ln 2[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572232044\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572232048\" class=\"exercise\">\n<div id=\"fs-id1170572232051\" class=\"textbox\">\n<p id=\"fs-id1170572232053\">Solve [latex]e^{2x} \/ (3+e^{2x})=1\/2[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572174654\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572174654\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572174654\">[latex]x=\\frac{\\ln 3}{2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165042579088\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042579095\">First solve the equation for [latex]e^{2x}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572174674\" class=\"textbox examples\">\n<h3>Solving Equations Involving Logarithmic Functions<\/h3>\n<div id=\"fs-id1170572174679\" class=\"exercise\">\n<div id=\"fs-id1170572174682\" class=\"textbox\">\n<p id=\"fs-id1170572174684\">Solve each of the following equations for [latex]x[\/latex].<\/p>\n<ol id=\"fs-id1170572174692\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\ln (\\frac{1}{x})=4[\/latex]<\/li>\n<li>[latex]\\log_{10} \\sqrt{x}+ \\log_{10} x=2[\/latex]<\/li>\n<li>[latex]\\ln(2x)-3 \\ln(x^2)=0[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572174799\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572174799\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572174799\" style=\"list-style-type: lower-alpha\">\n<li>By the definition of the natural logarithm function,\n<div id=\"fs-id1170573425282\" class=\"equation unnumbered\">[latex]\\ln\\big(\\frac{1}{x}\\big)=4 \\, \\text{if and only if} \\, e^4=\\frac{1}{x}[\/latex].<\/div>\n<p>Therefore, the solution is [latex]x=1\/e^4[\/latex].<\/li>\n<li>Using the product and power properties of logarithmic functions, rewrite the left-hand side of the equation as\n<div id=\"fs-id1170573416245\" class=\"equation unnumbered\">[latex]\\log_{10} \\sqrt{x}+ \\log_{10} x = \\log_{10} x \\sqrt{x} = \\log_{10}x^{3\/2} = \\frac{3}{2} \\log_{10} x[\/latex].<\/div>\n<p>Therefore, the equation can be rewritten as<\/p>\n<div id=\"fs-id1170571053549\" class=\"equation unnumbered\">[latex]\\frac{3}{2} \\log_{10} x = 2 \\, \\text{or} \\, \\log_{10} x = \\frac{4}{3}[\/latex].<\/div>\n<p>The solution is [latex]x=10^{4\/3}=10\\sqrt[3]{10}[\/latex].<\/li>\n<li>Using the power property of logarithmic functions, we can rewrite the equation as [latex]\\ln(2x) - \\ln(x^6) = 0[\/latex].<br \/>\nUsing the quotient property, this becomes<\/p>\n<div id=\"fs-id1170573426389\" class=\"equation unnumbered\">[latex]\\ln\\big(\\frac{2}{x^5}\\big)=0[\/latex].<\/div>\n<p>Therefore, [latex]2\/x^5=1[\/latex], which implies [latex]x=\\sqrt[5]{2}[\/latex]. We should then check for any extraneous solutions.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572552646\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572552650\" class=\"exercise\">\n<div id=\"fs-id1170572552652\" class=\"textbox\">\n<p id=\"fs-id1170572552654\">Solve [latex]\\ln(x^3)-4 \\ln (x)=1[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572552698\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572552698\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572552698\">[latex]x=\\frac{1}{e}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165043161242\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165043161250\">First use the power property, then use the product property of logarithms.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572552715\">When evaluating a logarithmic function with a calculator, you may have noticed that the only options are [latex]\\log_{10}[\/latex] or log, called the <span class=\"no-emphasis\"><em>common logarithm<\/em><\/span>, or ln, which is the natural logarithm. However, exponential functions and logarithm functions can be expressed in terms of any desired base [latex]b[\/latex]. If you need to use a calculator to evaluate an expression with a different base, you can apply the change-of-base formulas first. Using this change of base, we typically write a given exponential or logarithmic function in terms of the natural exponential and natural logarithmic functions.<\/p>\n<div id=\"fs-id1170572552753\" class=\"textbox key-takeaways\">\n<h3>Rule: Change-of-Base Formulas<\/h3>\n<p id=\"fs-id1170572267920\">Let [latex]a>0, \\, b>0[\/latex], and [latex]a\\ne 1, \\, b\\ne 1[\/latex].<\/p>\n<ol id=\"fs-id1170572267962\">\n<li>[latex]a^x=b^{x \\log_b a}[\/latex] for any real number [latex]x[\/latex].<br \/>\nIf [latex]b=e[\/latex], this equation reduces to [latex]a^x=e^{x \\log_e a}=e^{x \\ln a}[\/latex].<\/li>\n<li>[latex]\\log_a x=\\frac{\\log_b x}{\\log_b a}[\/latex] for any real number [latex]x>0[\/latex].<br \/>\nIf [latex]b=e[\/latex], this equation reduces to [latex]\\log_a x=\\frac{\\ln x}{\\ln a}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170572219407\" class=\"bc-section section\">\n<h2>Proof<\/h2>\n<p id=\"fs-id1170572219413\">For the first change-of-base formula, we begin by making use of the power property of logarithmic functions. We know that for any base [latex]b>0, \\, b\\ne 1, \\, \\log_b (a^x)=x \\log_b a[\/latex]. Therefore,<\/p>\n<div id=\"fs-id1170572219472\" class=\"equation unnumbered\">[latex]b^{\\log_b(a^x)}=b^{x \\log_b a}[\/latex].<\/div>\n<p id=\"fs-id1170572219523\">In addition, we know that [latex]b^x[\/latex] and [latex]\\log_b (x)[\/latex] are inverse functions. Therefore,<\/p>\n<div id=\"fs-id1170572219556\" class=\"equation unnumbered\">[latex]b^{\\log_b (a^x)}=a^x[\/latex].<\/div>\n<p id=\"fs-id1170572309868\">Combining these last two equalities, we conclude that [latex]a^x=b^{x \\log_b a}[\/latex].<\/p>\n<p id=\"fs-id1170572309898\">To prove the second property, we show that<\/p>\n<div id=\"fs-id1170572309902\" class=\"equation unnumbered\">[latex](\\log_b a)\u00b7(\\log_a x)=\\log_b x[\/latex].<\/div>\n<p id=\"fs-id1170572309957\">Let [latex]u=\\log_b a, \\, v=\\log_a x[\/latex], and [latex]w=\\log_b x[\/latex]. We will show that [latex]u\u00b7v=w[\/latex]. By the definition of logarithmic functions, we know that [latex]b^u=a, \\, a^v=x[\/latex], and [latex]b^w=x[\/latex]. From the previous equations, we see that<\/p>\n<div id=\"fs-id1170572434870\" class=\"equation unnumbered\">[latex]b^{uv}=(b^u)^v=a^v=x=b^w[\/latex].<\/div>\n<p id=\"fs-id1170572434927\">Therefore, [latex]b^{uv}=b^w[\/latex]. Since exponential functions are one-to-one, we can conclude that [latex]u\u00b7v=w[\/latex].<\/p>\n<p id=\"fs-id1170572434967\">\u25a1<\/p>\n<div id=\"fs-id1170572434970\" class=\"textbox examples\">\n<h3>Changing Bases<\/h3>\n<div id=\"fs-id1170572434975\" class=\"exercise\">\n<div id=\"fs-id1170572434977\" class=\"textbox\">\n<p id=\"fs-id1170572434979\">Use a calculating utility to evaluate [latex]\\log_3 7[\/latex] with the change-of-base formula presented earlier.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572435001\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572435001\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572435001\">Use the second equation with [latex]a=3[\/latex] and [latex]e=3[\/latex]:<\/p>\n<p id=\"fs-id1170572435025\">[latex]\\log_3 7=\\frac{\\ln 7}{\\ln 3} \\approx 1.77124[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572435059\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572435063\" class=\"exercise\">\n<div id=\"fs-id1170572435065\" class=\"textbox\">\n<p id=\"fs-id1170572435068\">Use the change-of-base formula and a calculating utility to evaluate [latex]\\log_4 6[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572128662\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572128662\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572128662\">1.29248<\/p>\n<\/div>\n<div id=\"fs-id1165043161231\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042853660\">Use the change of base to rewrite this expression in terms of expressions involving the natural logarithm function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572128672\" class=\"textbox examples\">\n<h3>Chapter Opener: The Richter Scale for Earthquakes<\/h3>\n<div id=\"fs-id1170572128677\" class=\"exercise\">\n<div id=\"fs-id1170572128679\" class=\"textbox\">\n<div id=\"CNX_Calc_Figure_01_05_010\" class=\"wp-caption aligncenter\">\n<div style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202656\/CNX_Calc_Figure_01_05_013.jpg\" alt=\"A photograph of an earthquake fault.\" width=\"325\" height=\"111\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 6: (credit: modification of work by Robb Hannawacker, NPS)<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572128701\">In 1935, Charles Richter developed a scale (now known as the <span class=\"no-emphasis\"><em>Richter scale<\/em><\/span>) to measure the magnitude of an <span class=\"no-emphasis\">earthquake<\/span>. The scale is a base-10 logarithmic scale, and it can be described as follows: Consider one earthquake with magnitude [latex]R_1[\/latex] on the Richter scale and a second earthquake with magnitude [latex]R_2[\/latex] on the Richter scale. Suppose [latex]R_1 > R_2[\/latex], which means the earthquake of magnitude [latex]R_1[\/latex] is stronger, but how much stronger is it than the other earthquake? A way of measuring the intensity of an earthquake is by using a seismograph to measure the amplitude of the earthquake waves. If [latex]A_1[\/latex] is the amplitude measured for the first earthquake and [latex]A_2[\/latex] is the amplitude measured for the second earthquake, then the amplitudes and magnitudes of the two earthquakes satisfy the following equation:<\/p>\n<div id=\"fs-id1170572128784\" class=\"equation unnumbered\">[latex]R_1 - R_2 = \\log_{10}(\\frac{A_1}{A_2})[\/latex].<\/div>\n<p id=\"fs-id1170572128836\">Consider an earthquake that measures 8 on the Richter scale and an earthquake that measures 7 on the Richter scale. Then,<\/p>\n<div id=\"fs-id1170572128840\" class=\"equation unnumbered\">[latex]8-7=\\log_{10}(\\frac{A_1}{A_2})[\/latex].<\/div>\n<p id=\"fs-id1170572233764\">Therefore,<\/p>\n<div id=\"fs-id1170572233767\" class=\"equation unnumbered\">[latex]\\log_{10}(\\frac{A_1}{A_2})=1[\/latex],<\/div>\n<p id=\"fs-id1170572233809\">which implies [latex]A_1 \/ A_2 = 10[\/latex] or [latex]A_1 = 10A_2[\/latex]. Since [latex]A_1[\/latex] is 10 times the size of [latex]A_2[\/latex], we say that the first earthquake is 10 times as intense as the second earthquake. On the other hand, if one earthquake measures 8 on the Richter scale and another measures 6, then the relative intensity of the two earthquakes satisfies the equation<\/p>\n<div id=\"fs-id1170572233874\" class=\"equation unnumbered\">[latex]\\log_{10}(\\frac{A_1}{A_2})=8-6=2[\/latex].<\/div>\n<p id=\"fs-id1170572233924\">Therefore, [latex]A_1=100A_2[\/latex]. That is, the first earthquake is 100 times more intense than the second earthquake.<\/p>\n<p id=\"fs-id1170572477962\">How can we use logarithmic functions to compare the relative severity of the magnitude 9 earthquake in Japan in 2011 with the magnitude 7.3 earthquake in Haiti in 2010?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572477974\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572477974\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572477974\">To compare the Japan and Haiti earthquakes, we can use an equation presented earlier:<\/p>\n<p id=\"fs-id1170572477977\">[latex]9-7.3=\\log_{10}(\\frac{A_1}{A_2})[\/latex].<\/p>\n<p id=\"fs-id1170572478019\">Therefore, [latex]A_1 \/ A_2=10^{1.7}[\/latex], and we conclude that the earthquake in Japan was approximately 50 times more intense than the earthquake in Haiti.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572478059\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572478063\" class=\"exercise\">\n<div id=\"fs-id1170572478065\" class=\"textbox\">\n<p id=\"fs-id1170572478067\">Compare the relative severity of a magnitude 8.4 earthquake with a magnitude 7.4 earthquake.<\/p>\n<\/div>\n<div class=\"solution\">\n<p id=\"fs-id1170572478135\">The magnitude 8.4 earthquake is roughly 10 times as severe as the magnitude 7.4 earthquake.<\/p>\n<\/div>\n<div id=\"fs-id1165039564589\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165039564597\">[latex]R_1-R_2=\\log_{10}(A_1 \/ A_2)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572478160\" class=\"bc-section section\">\n<h1>Hyperbolic Functions<\/h1>\n<p id=\"fs-id1170572467918\">The <strong>hyperbolic functions<\/strong> are defined in terms of certain combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]. These functions arise naturally in various engineering and physics applications, including the study of water waves and vibrations of elastic membranes. Another common use for a hyperbolic function is the representation of a hanging chain or cable, also known as a catenary (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_006\">(Figure)<\/a>). If we introduce a coordinate system so that the low point of the chain lies along the [latex]y[\/latex]-axis, we can describe the height of the chain in terms of a hyperbolic function. First, we define the hyperbolic functions.<\/p>\n<div id=\"CNX_Calc_Figure_01_05_006\" class=\"wp-caption aligncenter\">\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202700\/CNX_Calc_Figure_01_05_009.jpg\" alt=\"A photograph of a spider web collecting dew drops.\" width=\"488\" height=\"403\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 7. The shape of a strand of silk in a spider\u2019s web can be described in terms of a hyperbolic function. The same shape applies to a chain or cable hanging from two supports with only its own weight. (credit: \u201cMtpaley\u201d, Wikimedia Commons)<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572467985\" class=\"textbox key-takeaways\">\n<div class=\"title\">\n<h3>Definition<\/h3>\n<\/div>\n<p id=\"fs-id1170572467989\"><strong>Hyperbolic cosine<\/strong><\/p>\n<div id=\"fs-id1170572467996\" class=\"equation unnumbered\">[latex]\\cosh x=\\large \\frac{e^x+e^{\u2212x}}{2}[\/latex]<\/div>\n<p id=\"fs-id1170572468030\"><strong>Hyperbolic sine<\/strong><\/p>\n<div id=\"fs-id1170572468036\" class=\"equation unnumbered\">[latex]\\sinh x=\\large \\frac{e^x-e^{\u2212x}}{2}[\/latex]<\/div>\n<p id=\"fs-id1170572468070\"><strong>Hyperbolic tangent<\/strong><\/p>\n<div id=\"fs-id1170572468077\" class=\"equation unnumbered\">[latex]\\tanh x=\\large \\frac{\\sinh x}{\\cosh x} \\normalsize = \\large \\frac{e^x-e^{\u2212x}}{e^x+e^{\u2212x}}[\/latex]<\/div>\n<p id=\"fs-id1170572431436\"><strong>Hyperbolic cosecant<\/strong><\/p>\n<div id=\"fs-id1170572431443\" class=\"equation unnumbered\">[latex]\\text{csch} \\, x=\\large \\frac{1}{\\sinh x} \\normalsize = \\large \\frac{2}{e^x-e^{\u2212x}}[\/latex]<\/div>\n<p id=\"fs-id1170572431488\"><strong>Hyperbolic secant<\/strong><\/p>\n<div id=\"fs-id1170572431494\" class=\"equation unnumbered\">[latex]\\text{sech} \\, x=\\large \\frac{1}{\\cosh x} \\normalsize = \\large \\frac{2}{e^x+e^{\u2212x}}[\/latex]<\/div>\n<p id=\"fs-id1170572431539\"><strong>Hyperbolic cotangent<\/strong><\/p>\n<div id=\"fs-id1170572431545\" class=\"equation unnumbered\">[latex]\\coth x=\\large \\frac{\\cosh x}{\\sinh x} \\normalsize = \\large \\frac{e^x+e^{\u2212x}}{e^x-e^{\u2212x}}[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1170572431609\">The name <em>cosh<\/em> rhymes with \u201cgosh,\u201d whereas the name <em>sinh<\/em> is pronounced \u201ccinch.\u201d <em>Tanh<\/em>, <em>sech<\/em>, <em>csch<\/em>, and <em>coth<\/em> are pronounced \u201ctanch,\u201d \u201cseech,\u201d \u201ccoseech,\u201d and \u201ccotanch,\u201d respectively.<\/p>\n<p id=\"fs-id1170572234293\">Using the definition of [latex]\\cosh(x)[\/latex] and principles of physics, it can be shown that the height of a hanging chain, such as the one in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_006\">(Figure)<\/a>, can be described by the function [latex]h(x)=a \\cosh(x\/a)+c[\/latex] for certain constants [latex]a[\/latex] and [latex]c[\/latex].<\/p>\n<p id=\"fs-id1170572234364\">But why are these functions called <em>hyperbolic functions<\/em>? To answer this question, consider the quantity [latex]\\cosh^2 t-\\sinh^2 t[\/latex]. Using the definition of [latex]\\cosh[\/latex] and [latex]\\sinh[\/latex], we see that<\/p>\n<div id=\"fs-id1170572234411\" class=\"equation unnumbered\">[latex]\\cosh^2 t-\\sinh^2 t=\\large \\frac{e^{2t}+2+e^{-2t}}{4}-\\frac{e^{2t}-2+e^{-2t}}{4} \\normalsize =1[\/latex].<\/div>\n<p id=\"fs-id1170572171610\">This identity is the analog of the trigonometric identity [latex]\\cos^2 t+\\sin^2 t=1[\/latex]. Here, given a value [latex]t[\/latex], the point [latex](x,y)=(\\cosh t,\\sinh t)[\/latex] lies on the unit hyperbola [latex]x^2-y^2=1[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_007\">(Figure)<\/a>).<\/p>\n<div id=\"CNX_Calc_Figure_01_05_007\" class=\"wp-caption aligncenter\">\n<div style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202702\/CNX_Calc_Figure_01_05_007.jpg\" alt=\"An image of a graph. The x axis runs from -1 to 3 and the y axis runs from -3 to 3. The graph is of the relation \u201c(x squared) - (y squared) -1\u201d. The left most point of the relation is at the x intercept, which is at the point (1, 0). From this point the relation both increases and decreases in curves as x increases. This relation is known as a hyperbola and it resembles a sideways \u201cU\u201d shape. There is a point plotted on the graph of the relation labeled \u201c(cosh(1), sinh(1))\u201d, which is at the approximate point (1.5, 1.2).\" width=\"325\" height=\"275\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 8.The unit hyperbola [latex]\\cosh^2 t-\\sinh^2 t=1[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572171758\" class=\"bc-section section\">\n<h2>Graphs of Hyperbolic Functions<\/h2>\n<p id=\"fs-id1170572171763\">To graph [latex]\\cosh x[\/latex] and [latex]\\sinh x[\/latex], we make use of the fact that both functions approach [latex](1\/2)e^x[\/latex] as [latex]x \\to \\infty[\/latex], since [latex]e^{\u2212x} \\to 0[\/latex] as [latex]x \\to \\infty[\/latex]. As [latex]x \\to \u2212\\infty, \\, \\cosh x[\/latex] approaches [latex]1\/2e^{\u2212x}[\/latex], whereas [latex]\\sinh x[\/latex] approaches [latex]-1\/2e^{\u2212x}[\/latex]. Therefore, using the graphs of [latex]1\/2e^x, \\, 1\/2e^{\u2212x}[\/latex], and [latex]\u22121\/2e^{\u2212x}[\/latex] as guides, we graph [latex]\\cosh x[\/latex] and [latex]\\sinh x[\/latex]. To graph [latex]\\tanh x[\/latex], we use the fact that [latex]\\tanh(0)=1, \\, -1<\\tanh(x)<1[\/latex] for all [latex]x, \\, \\tanh x \\to 1[\/latex] as [latex]x \\to \\infty[\/latex], and [latex]\\tanh x \\to \u22121[\/latex] as [latex]x \\to \u2212\\infty[\/latex]. The graphs of the other three hyperbolic functions can be sketched using the graphs of [latex]\\cosh x, \\, \\sinh x[\/latex], and [latex]\\tanh x[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_008\">(Figure)<\/a>).<\/p>\n<div id=\"CNX_Calc_Figure_01_05_008\" class=\"wp-caption aligncenter\">\n<div style=\"width: 583px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202706\/CNX_Calc_Figure_01_05_011.jpg\" alt=\"An image of six graphs. Each graph has an x axis that runs from -3 to 3 and a y axis that runs from -4 to 4. The first graph is of the function \u201cy = cosh(x)\u201d, which is a hyperbola. The function decreases until it hits the point (0, 1), where it begins to increase. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = (1\/2)(e to power of -x)\u201d, a decreasing curved function and the second of these functions is \u201cy = (1\/2)(e to power of x)\u201d, an increasing curved function. The function \u201cy = cosh(x)\u201d is always above these two functions without ever touching them. The second graph is of the function \u201cy = sinh(x)\u201d, which is an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = (1\/2)(e to power of x)\u201d, an increasing curved function and the second of these functions is \u201cy = -(1\/2)(e to power of -x)\u201d, an increasing curved function that approaches the x axis without touching it. The function \u201cy = sinh(x)\u201d is always between these two functions without ever touching them. The third graph is of the function \u201cy = sech(x)\u201d, which increases until the point (0, 1), where it begins to decrease. The graph of the function has a hump. The fourth graph is of the function \u201cy = csch(x)\u201d. On the left side of the y axis, the function starts slightly below the x axis and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the x axis, which it never touches. The fifth graph is of the function \u201cy = tanh(x)\u201d, an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = 1\u201d, a horizontal line function and the second of these functions is \u201cy = -1\u201d, another horizontal line function. The function \u201cy = tanh(x)\u201d is always between these two functions without ever touching them. The sixth graph is of the function \u201cy = coth(x)\u201d. On the left side of the y axis, the function starts slightly below the boundary line \u201cy = 1\u201d and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the boundary line \u201cy = -1\u201d, which it never touches.\" width=\"573\" height=\"929\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 9. The hyperbolic functions involve combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572433248\" class=\"bc-section section\">\n<h2>Identities Involving Hyperbolic Functions<\/h2>\n<p id=\"fs-id1170572433254\">The identity [latex]\\cosh^2 t-\\sinh^2 t[\/latex], shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_01_05_007\">(Figure)<\/a>, is one of several identities involving the hyperbolic functions, some of which are listed next. The first four properties follow easily from the definitions of hyperbolic sine and hyperbolic cosine. Except for some differences in signs, most of these properties are analogous to identities for trigonometric functions.<\/p>\n<div id=\"fs-id1170572433288\" class=\"textbox key-takeaways\">\n<h3>Rule: Identities Involving Hyperbolic Functions<\/h3>\n<ol id=\"fs-id1170572433294\">\n<li>[latex]\\cosh(\u2212x)=\\cosh x[\/latex]<\/li>\n<li>[latex]\\sinh(\u2212x)=\u2212\\sinh x[\/latex]<\/li>\n<li>[latex]\\cosh x+\\sinh x=e^x[\/latex]<\/li>\n<li>[latex]\\cosh x-\\sinh x=e^{\u2212x}[\/latex]<\/li>\n<li>[latex]\\cosh^2 x-\\sinh^2 x=1[\/latex]<\/li>\n<li>[latex]1-\\tanh^2 x=\\text{sech}^2 x[\/latex]<\/li>\n<li>[latex]\\coth^2 x-1=\\text{csch}^2 x[\/latex]<\/li>\n<li>[latex]\\sinh(x \\pm y)=\\sinh x \\cosh y \\pm \\cosh x \\sinh y[\/latex]<\/li>\n<li>[latex]\\cosh (x \\pm y)=\\cosh x \\cosh y \\pm \\sinh x \\sinh y[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170572443393\" class=\"textbox examples\">\n<h3>Evaluating Hyperbolic Functions<\/h3>\n<div id=\"fs-id1170572443399\" class=\"exercise\">\n<div id=\"fs-id1170572443401\" class=\"textbox\">\n<ol id=\"fs-id1170572443403\" style=\"list-style-type: lower-alpha\">\n<li>Simplify [latex]\\sinh(5 \\ln x)[\/latex].<\/li>\n<li>If [latex]\\sinh x=3\/4[\/latex], find the values of the remaining five hyperbolic functions.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572443462\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572443462\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572443462\" style=\"list-style-type: lower-alpha\">\n<li>Using the definition of the [latex]\\sinh[\/latex] function, we write\n<div id=\"fs-id1170570995857\" class=\"equation unnumbered\">[latex]\\sinh(5 \\ln x)=\\large \\frac{e^{5 \\ln x}-e^{-5 \\ln x}}{2} \\normalsize = \\large \\frac{e^{\\ln(x^5)}-e^{\\ln(x^{-5})}}{2} \\normalsize =\\large \\frac{x^5-x^{-5}}{2}[\/latex].<\/div>\n<\/li>\n<li>Using the identity [latex]\\cosh^2 x-\\sinh^2 x=1[\/latex], we see that\n<div id=\"fs-id1170573388429\" class=\"equation unnumbered\">[latex]\\cosh^2 x=1+\\big(\\frac{3}{4}\\big)^2=\\frac{25}{16}[\/latex].<\/div>\n<p>Since [latex]\\cosh x \\ge 1[\/latex] for all [latex]x[\/latex], we must have [latex]\\cosh x=5\/4[\/latex]. Then, using the definitions for the other hyperbolic functions, we conclude that [latex]\\tanh x=3\/5, \\, \\text{csch} \\, x=4\/3, \\, \\text{sech} \\, x=4\/5[\/latex], and [latex]\\coth x=5\/3[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572548314\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572548318\" class=\"exercise\">\n<div class=\"textbox\">\n<p id=\"fs-id1170572549916\">Simplify [latex]\\cosh(2 \\ln x)[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572549946\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572549946\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572549946\">[latex](x^2+x^{-2})\/2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165039563328\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165039563336\">Use the definition of the cosh function and the power property of logarithm functions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572549981\" class=\"bc-section section\">\n<h2>Inverse Hyperbolic Functions<\/h2>\n<p id=\"fs-id1170572549987\">From the graphs of the hyperbolic functions, we see that all of them are one-to-one except [latex]\\cosh x[\/latex] and [latex]\\text{sech} \\, x[\/latex]. If we restrict the domains of these two functions to the interval [latex][0,\\infty)[\/latex], then all the hyperbolic functions are one-to-one, and we can define the inverse hyperbolic functions. Since the hyperbolic functions themselves involve exponential functions, the inverse hyperbolic functions involve logarithmic functions.<\/p>\n<div id=\"fs-id1170572550032\" class=\"textbox key-takeaways\">\n<div class=\"title\">\n<h3>Definition<\/h3>\n<\/div>\n<p id=\"fs-id1170572550036\"><strong>Inverse Hyperbolic Functions<\/strong><\/p>\n<div id=\"fs-id1170572550043\" class=\"equation unnumbered\">[latex]\\begin{array}{cccc}\\sinh^{-1} x=\\text{arcsinh } x=\\ln(x+\\sqrt{x^2+1})\\hfill & & & \\cosh^{-1} x=\\text{arccosh } x=\\ln(x+\\sqrt{x^2-1})\\hfill \\\\ \\tanh^{-1} x=\\text{arctanh } x=\\frac{1}{2}\\ln\\big(\\frac{1+x}{1-x}\\big)\\hfill & & & \\coth^{-1} x=\\text{arccot } x=\\frac{1}{2}\\ln\\big(\\frac{x+1}{x-1}\\big)\\hfill \\\\ \\text{sech}^{-1} x=\\text{arcsech } x=\\ln\\big(\\frac{1+\\sqrt{1-x^2}}{x}\\big)\\hfill & & & \\text{csch}^{-1} x=\\text{arccsch } x=\\ln\\big(\\frac{1}{x}+\\frac{\\sqrt{1+x^2}}{|x|}\\big)\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1170572235136\">Let\u2019s look at how to derive the first equation. The others follow similarly. Suppose [latex]y=\\sinh^{-1} x[\/latex]. Then, [latex]x=\\sinh y[\/latex] and, by the definition of the hyperbolic sine function, [latex]x=\\frac{e^y-e^{\u2212y}}{2}[\/latex]. Therefore,<\/p>\n<div id=\"fs-id1170572235202\" class=\"equation unnumbered\">[latex]e^y-2x-e^{\u2212y}=0[\/latex].<\/div>\n<p>Multiplying this equation by [latex]e^y[\/latex], we obtain<\/p>\n<div class=\"equation unnumbered\">[latex]e^{2y}-2xe^y-1=0[\/latex].<\/div>\n<p id=\"fs-id1170572235286\">This can be solved like a quadratic equation, with the solution<\/p>\n<div id=\"fs-id1170572235290\" class=\"equation unnumbered\">[latex]e^y=\\large \\frac{2x \\pm \\sqrt{4x^2+4}}{2} \\normalsize =x \\pm \\sqrt{x^2+1}[\/latex].<\/div>\n<p id=\"fs-id1170572482176\">Since [latex]e^y>0[\/latex], the only solution is the one with the positive sign. Applying the natural logarithm to both sides of the equation, we conclude that<\/p>\n<div id=\"fs-id1170572482196\" class=\"equation unnumbered\">[latex]y=\\ln(x+\\sqrt{x^2+1})[\/latex].<\/div>\n<div id=\"fs-id1170572482234\" class=\"textbox examples\">\n<h3>Evaluating Inverse Hyperbolic Functions<\/h3>\n<div id=\"fs-id1170572482240\" class=\"exercise\">\n<div id=\"fs-id1170572482242\" class=\"textbox\">\n<p id=\"fs-id1170572482244\">Evaluate each of the following expressions.<\/p>\n<div id=\"fs-id1170572482247\" class=\"equation unnumbered\">[latex]\\sinh^{-1}(2)[\/latex]<\/div>\n<div id=\"fs-id1170572482270\" class=\"equation unnumbered\">[latex]\\tanh^{-1}(1\/4)[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572482306\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572482306\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572482306\">[latex]\\sinh^{-1}(2)=\\ln(2+\\sqrt{2^2+1})=\\ln(2+\\sqrt{5}) \\approx 1.4436[\/latex]<\/p>\n<p id=\"fs-id1170572176029\">[latex]\\tanh^{-1}(1\/4)=\\frac{1}{2}\\ln(\\frac{1+1\/4}{1-1\/4})=\\frac{1}{2}\\ln(\\frac{5\/4}{3\/4})=\\frac{1}{2}\\ln(\\frac{5}{3}) \\approx 0.2554[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572176157\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572176161\" class=\"exercise\">\n<div id=\"fs-id1170572176163\" class=\"textbox\">\n<p id=\"fs-id1170572176165\">Evaluate [latex]\\tanh^{-1}(1\/2)[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572176215\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572176215\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572176215\">[latex]\\frac{1}{2}\\ln(3) \\approx 0.5493[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165039559588\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165039559595\">Use the definition of [latex]\\tanh^{-1} x[\/latex] and simplify.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572176246\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1170572216270\">\n<li>The exponential function [latex]y=b^x[\/latex] is increasing if [latex]b>1[\/latex] and decreasing if [latex]0<b<1[\/latex]. Its domain is [latex](\u2212\\infty ,\\infty)[\/latex] and its range is [latex](0,\\infty)[\/latex].<\/li>\n<li>The logarithmic function [latex]y=\\log_b(x)[\/latex] is the inverse of [latex]y=b^x[\/latex]. Its domain is [latex](0,\\infty)[\/latex] and its range is [latex](\u2212\\infty,\\infty)[\/latex].<\/li>\n<li>The natural exponential function is [latex]y=e^x[\/latex] and the natural logarithmic function is [latex]y=\\ln x=\\log_e x[\/latex].<\/li>\n<li>Given an exponential function or logarithmic function in base [latex]a[\/latex], we can make a change of base to convert this function to any base [latex]b>0, \\, b \\ne 1[\/latex]. We typically convert to base [latex]e[\/latex].<\/li>\n<li>The hyperbolic functions involve combinations of the exponential functions [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]. As a result, the inverse hyperbolic functions involve the natural logarithm.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1170572545884\" class=\"textbox exercises\">\n<p id=\"fs-id1170572545888\">For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal.<\/p>\n<div id=\"fs-id1170572545893\" class=\"exercise\">\n<div id=\"fs-id1170572545895\" class=\"textbox\">\n<p id=\"fs-id1170572545897\"><strong>1.\u00a0<\/strong>[latex]f(x)=5^x[\/latex] a. [latex]x=3[\/latex] b. [latex]x=\\frac{1}{2}[\/latex] c. [latex]x=\\sqrt{2}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572545955\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572545955\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572545955\">a. 125 b. 2.24 c. 9.74<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572545960\" class=\"exercise\">\n<div id=\"fs-id1170572545962\" class=\"textbox\">\n<p id=\"fs-id1170572545965\"><strong>2.\u00a0<\/strong>[latex]f(x)=(0.3)^x[\/latex] a. [latex]x=-1[\/latex] b. [latex]x=4[\/latex] c. [latex]x=-1.5[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572546032\" class=\"exercise\">\n<div id=\"fs-id1170572546034\" class=\"textbox\">\n<p id=\"fs-id1170572546036\"><strong>3.\u00a0<\/strong>[latex]f(x)=10^x[\/latex] a. [latex]x=-2[\/latex] b. [latex]x=4[\/latex] c. [latex]x=\\frac{5}{3}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572358150\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572358150\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572358150\">a. 0.01 b. 10,000 c. 46.42<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572358156\" class=\"exercise\">\n<div id=\"fs-id1170572358158\" class=\"textbox\">\n<p id=\"fs-id1170572358160\"><strong>4.\u00a0<\/strong>[latex]f(x)=e^x[\/latex] a. [latex]x=2[\/latex] b. [latex]x=-3.2[\/latex] c. [latex]x=\\pi[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572358219\">For the following exercises, match the exponential equation to the correct graph.<\/p>\n<ol id=\"fs-id1170572358222\" style=\"list-style-type: lower-alpha\">\n<li>[latex]y=4^{\u2212x}[\/latex]<\/li>\n<li>[latex]y=3^{x-1}[\/latex]<\/li>\n<li>[latex]y=2^{x+1}[\/latex]<\/li>\n<li>[latex]y=(\\frac{1}{2})^x+2[\/latex]<\/li>\n<li>[latex]y=\u22123^{\u2212x}[\/latex]<\/li>\n<li>[latex]y=1-5^x[\/latex]<\/li>\n<\/ol>\n<div id=\"fs-id1170572358351\" class=\"exercise\">\n<div id=\"fs-id1170572358353\" class=\"textbox\"><span id=\"fs-id1170572358359\"><strong>5.\u00a0<\/strong><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202709\/CNX_Calc_Figure_01_05_201.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -2 to 8. The graph is of a decreasing curved function. The function decreases until it approaches the line \u201cy = 2\u201d, but never touches this line. The y intercept is at the point (0, 3) and there is no x intercept.\" \/><\/span><\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572553700\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572553700\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572553700\">d<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572553705\" class=\"exercise\">\n<div id=\"fs-id1170572553707\" class=\"textbox\"><span id=\"fs-id1170572553713\"><strong>6.\u00a0<\/strong><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202713\/CNX_Calc_Figure_01_05_202.jpg\" alt=\"An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -9 to 2. The graph is of a function that starts slightly below the line \u201cy = 1\u201d and begins decreasing rapidly in a curve. The x intercept and y intercept are both at the origin.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1170572553732\" class=\"exercise\">\n<div id=\"fs-id1170572553734\" class=\"textbox\"><span id=\"fs-id1170572553737\"><strong>7.\u00a0<\/strong><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202716\/CNX_Calc_Figure_01_05_203.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that starts slightly above the x axis and begins increasing rapidly. There is no x intercept and the y intercept is at the point (0, (1\/3)). Another point of the graph is at (1, 1).\" \/><\/span><\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572553754\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572553754\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572553754\">b<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572553760\" class=\"exercise\">\n<div id=\"fs-id1170572553762\" class=\"textbox\"><span id=\"fs-id1170572553764\"><strong>8.\u00a0<\/strong><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202719\/CNX_Calc_Figure_01_05_204.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved decreasing function that decreases until it comes close the x axis without touching it. There is no x intercept and the y intercept is at the point (0, 1). Another point of the graph is at (-1, 4).\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1170572553787\" class=\"exercise\">\n<div id=\"fs-id1170572553789\" class=\"textbox\"><span id=\"fs-id1170572553791\"><strong>9.\u00a0<\/strong><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202722\/CNX_Calc_Figure_01_05_205.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that increases until it comes close the x axis without touching it. There is no x intercept and the y intercept is at the point (0, -1). Another point of the graph is at (-1, -3).\" \/><\/span><\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572553809\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572553809\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572553809\">e<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572553814\" class=\"exercise\">\n<div id=\"fs-id1170572553816\" class=\"textbox\"><span id=\"fs-id1170572553818\"><strong>10.<\/strong><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202725\/CNX_Calc_Figure_01_05_206.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that starts slightly above the x axis and begins increasing rapidly. There is no x intercept and the y intercept is at the point (0, 2). Another point of the graph is at (-1, 1).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170572553841\">For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote.<\/p>\n<div id=\"fs-id1170572553846\" class=\"exercise\">\n<div id=\"fs-id1170572553848\" class=\"textbox\">\n<p id=\"fs-id1170572553850\"><strong>11.\u00a0<\/strong>[latex]f(x)=e^x+2[\/latex]<\/p>\n<p><span id=\"fs-id1170572553879\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202729\/CNX_Calc_Figure_01_05_207.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that starts slightly above the line \u201cy = 2\u201d and begins increasing rapidly. There is no x intercept and the y intercept is at the point (0, 3).\" \/><\/span><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572553893\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572553893\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572553893\">Domain: all real numbers, Range: [latex](2,\\infty)[\/latex], Horizontal asymptote at [latex]y=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572553923\" class=\"exercise\">\n<div id=\"fs-id1170572553925\" class=\"textbox\">\n<p id=\"fs-id1170572553927\"><strong>12.\u00a0<\/strong>[latex]f(x)=\u22122^x[\/latex]<\/p>\n<p><span id=\"fs-id1170572242925\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202731\/CNX_Calc_Figure_01_05_208.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a function that starts slightly below the x axis and begins decreasing rapidly in a curve. There is no x intercept and y intercept is at the point (0, -1).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572242973\" class=\"exercise\">\n<div id=\"fs-id1170572242975\" class=\"textbox\">\n<p id=\"fs-id1170572242977\"><strong>13.\u00a0<\/strong>[latex]f(x)=3^{x+1}[\/latex]<\/p>\n<p><span id=\"fs-id1170572243004\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202734\/CNX_Calc_Figure_01_05_209.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that starts slightly above the x axis and begins increasing rapidly. There is no x intercept and the y intercept is at the point (0, 3). Another point of the graph is at (-1, 1).\" \/><\/span><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572243022\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572243022\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572243022\">Domain: all real numbers, Range: [latex](0,\\infty)[\/latex], Horizontal asymptote at [latex]y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572243051\" class=\"exercise\">\n<div id=\"fs-id1170572243053\" class=\"textbox\">\n<p id=\"fs-id1170572243055\"><strong>14.\u00a0<\/strong>[latex]f(x)=4^x-1[\/latex]<\/p>\n<p><span id=\"fs-id1170572243084\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202738\/CNX_Calc_Figure_01_05_210.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that starts slightly above the line \u201cy = -1\u201d and begins increasing rapidly. There x intercept and the y intercept are both at the origin. Another point of the graph is at (1, 3).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572243129\" class=\"exercise\">\n<div id=\"fs-id1170572243131\" class=\"textbox\">\n<p id=\"fs-id1170572243133\"><strong>15.\u00a0<\/strong>[latex]f(x)=1-2^{\u2212x}[\/latex]<\/p>\n<p><span id=\"fs-id1170572163713\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202741\/CNX_Calc_Figure_01_05_211.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved increasing function that increases until it comes close the line \u201cy = 1\u201d without touching it. There x intercept and the y intercept are both at the origin. Another point of the graph is at (-1, -1).\" \/><\/span><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572163728\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572163728\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572163728\">Domain: all real numbers, Range: [latex](\u2212\\infty ,1)[\/latex], Horizontal asymptote at [latex]y=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572163759\" class=\"exercise\">\n<div id=\"fs-id1170572163761\" class=\"textbox\">\n<p id=\"fs-id1170572163763\"><strong>16.\u00a0<\/strong>[latex]f(x)=5^{x+1}+2[\/latex]<\/p>\n<p><span id=\"fs-id1170572163798\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202744\/CNX_Calc_Figure_01_05_212.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -1 to 9. The graph is of a curved increasing function that starts slightly above the line \u201cy = 2\u201d and begins increasing rapidly. There is no x intercept and the y intercept is at the point (0, 7). Another point of the graph is at (-1, 3).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572163842\" class=\"exercise\">\n<div id=\"fs-id1170572163844\" class=\"textbox\">\n<p id=\"fs-id1170572163846\"><strong>17.\u00a0<\/strong>[latex]f(x)=e^{\u2212x}-1[\/latex]<\/p>\n<p><span id=\"fs-id1170572163879\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202748\/CNX_Calc_Figure_01_05_213.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a curved decreasing function that decreases until it comes close the line \u201cy = -1\u201d without touching it. There x intercept and the y intercept are both at the origin. There is an approximate point on the graph at (-1, 1.7).\" \/><\/span><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572163893\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572163893\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572163893\">Domain: all real numbers, Range: [latex](-1,\\infty )[\/latex], Horizontal asymptote at [latex]y=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572163923\">For the following exercises, write the equation in equivalent exponential form.<\/p>\n<div id=\"fs-id1170572163926\" class=\"exercise\">\n<div id=\"fs-id1170572163928\" class=\"textbox\">\n<p id=\"fs-id1170572163930\"><strong>18.\u00a0<\/strong>[latex]\\log_3 81=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572218452\" class=\"exercise\">\n<div id=\"fs-id1170572218455\" class=\"textbox\">\n<p id=\"fs-id1170572218457\"><strong>19.\u00a0<\/strong>[latex]\\log_8 2=\\frac{1}{3}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572218480\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572218480\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572218480\">[latex]8^{1\/3}=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572218502\" class=\"exercise\">\n<div id=\"fs-id1170572218504\" class=\"textbox\">\n<p id=\"fs-id1170572218506\"><strong>20.\u00a0<\/strong>[latex]\\log_5 1=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572218542\" class=\"exercise\">\n<div id=\"fs-id1170572218544\" class=\"textbox\">\n<p id=\"fs-id1170572218546\"><strong>21.\u00a0<\/strong>[latex]\\log_5 25=2[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572218567\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572218567\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572218567\">[latex]5^2=25[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572218583\" class=\"exercise\">\n<div id=\"fs-id1170572218585\" class=\"textbox\">\n<p id=\"fs-id1170572218587\"><strong>22.\u00a0<\/strong>[latex]\\log 0.1=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572218621\" class=\"exercise\">\n<div id=\"fs-id1170572218623\" class=\"textbox\">\n<p id=\"fs-id1170572218626\"><strong>23.\u00a0<\/strong>[latex]\\ln(\\frac{1}{e^3})=-3[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572218656\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572218656\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572218656\">[latex]e^{-3}=\\frac{1}{e^3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572168644\" class=\"exercise\">\n<div id=\"fs-id1170572168646\" class=\"textbox\">\n<p id=\"fs-id1170572168649\"><strong>24.\u00a0<\/strong>[latex]\\log_9 3=0.5[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572168686\" class=\"exercise\">\n<div id=\"fs-id1170572168688\" class=\"textbox\">\n<p id=\"fs-id1170572168690\"><strong>25.\u00a0<\/strong>[latex]\\ln 1=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572168706\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572168706\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572168706\">[latex]e^0=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572168722\">For the following exercises, write the equation in equivalent logarithmic form.<\/p>\n<div id=\"fs-id1170572168725\" class=\"exercise\">\n<div id=\"fs-id1170572168728\" class=\"textbox\">\n<p id=\"fs-id1170572168730\"><strong>26.\u00a0<\/strong>[latex]2^3=8[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572168766\" class=\"exercise\">\n<div id=\"fs-id1170572168768\" class=\"textbox\">\n<p id=\"fs-id1170572168770\"><strong>27.\u00a0<\/strong>[latex]4^{-2}=\\frac{1}{16}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572168793\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572168793\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572168793\">[latex]\\log_4(\\frac{1}{16})=-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572168823\" class=\"exercise\">\n<div id=\"fs-id1170572168825\" class=\"textbox\">\n<p id=\"fs-id1170572168827\"><strong>28.\u00a0<\/strong>[latex]10^2=100[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572168860\" class=\"exercise\">\n<div id=\"fs-id1170572168862\" class=\"textbox\">\n<p id=\"fs-id1170572168865\"><strong>29.\u00a0<\/strong>[latex]9^0=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572168882\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572168882\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572168882\">[latex]\\log_9 1=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572218179\" class=\"exercise\">\n<div id=\"fs-id1170572218181\" class=\"textbox\">\n<p id=\"fs-id1170572218184\"><strong>30.\u00a0<\/strong>[latex](\\frac{1}{3})^3=\\frac{1}{27}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572218251\" class=\"exercise\">\n<div id=\"fs-id1170572218253\" class=\"textbox\">\n<p id=\"fs-id1170572218255\"><strong>31.\u00a0<\/strong>[latex]\\sqrt[3]{64}=4[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572218273\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572218273\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572218273\">[latex]\\log_{64} 4=\\frac{1}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572218297\" class=\"exercise\">\n<div id=\"fs-id1170572218299\" class=\"textbox\">\n<p id=\"fs-id1170572218301\"><strong>32.\u00a0<\/strong>[latex]e^x=y[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572218333\" class=\"exercise\">\n<div id=\"fs-id1170572218335\" class=\"textbox\">\n<p id=\"fs-id1170572218337\"><strong>33.\u00a0<\/strong>[latex]9^y=150[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572218354\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572218354\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572218354\">[latex]\\log_9 150=y[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572218373\" class=\"exercise\">\n<div id=\"fs-id1170572218376\" class=\"textbox\">\n<p id=\"fs-id1170572218378\"><strong>34.\u00a0<\/strong>[latex]b^3=45[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572218414\" class=\"exercise\">\n<div id=\"fs-id1170572218416\" class=\"textbox\">\n<p id=\"fs-id1170572218418\"><strong>35.\u00a0<\/strong>[latex]4^{-3\/2}=0.125[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572217083\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572217083\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572217083\">[latex]\\log_4 0.125=-\\frac{3}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572217107\">For the following exercises, use the given graphs of the logarithmic functions to determine their domain, range, and vertical asymptote.<\/p>\n<div id=\"fs-id1170572217112\" class=\"exercise\">\n<div id=\"fs-id1170572217114\" class=\"textbox\">\n<p id=\"fs-id1170572217116\"><strong>36.\u00a0<\/strong>[latex]f(x)=3+\\ln x[\/latex]<\/p>\n<p><span id=\"fs-id1170572217140\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202751\/CNX_Calc_Figure_01_05_214.jpg\" alt=\"An image of a graph. The x axis runs from -10 to 10 and the y axis runs from -10 to 10. The graph is of an increasing curved function which starts slightly to the right of the y axis. There is no y intercept and the x intercept is at the approximate point (0.05, 0). Another point on the graph is (1, 3).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572217209\" class=\"exercise\">\n<div id=\"fs-id1170572217211\" class=\"textbox\">\n<p id=\"fs-id1170572217213\"><strong>37.\u00a0<\/strong>[latex]f(x)=\\ln(x-1)[\/latex]<\/p>\n<p><span id=\"fs-id1170572217249\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202754\/CNX_Calc_Figure_01_05_215.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of an increasing curved function which starts slightly to the right of the vertical line \u201cx = 1\u201d. There is no y intercept and the x intercept is at the approximate point (2, 0).\" \/><\/span><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572217264\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572217264\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572217264\">Domain: [latex](1,\\infty ),[\/latex] Range: [latex](\u2212\\infty ,\\infty)[\/latex], Vertical asymptote at [latex]x=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572217315\" class=\"exercise\">\n<div id=\"fs-id1170572217317\" class=\"textbox\">\n<p id=\"fs-id1170572286631\"><strong>38.\u00a0<\/strong>[latex]f(x)=\\ln(\u2212x)[\/latex]<\/p>\n<p><span id=\"fs-id1170572286662\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202758\/CNX_Calc_Figure_01_05_216.jpg\" alt=\"An image of a graph. The x axis runs from -9 to 1 and the y axis runs from -5 to 5. The graph is a curved decreasing function that approaches the y axis without touching it. There is no y intercept and the x intercept is at (-1, 0).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572286731\" class=\"exercise\">\n<div id=\"fs-id1170572286734\" class=\"textbox\">\n<p id=\"fs-id1170572286736\"><strong>39.\u00a0<\/strong>[latex]f(x)=1-\\ln x[\/latex]<\/p>\n<p><span id=\"fs-id1170572286760\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202801\/CNX_Calc_Figure_01_05_217.jpg\" alt=\"An image of a graph. The x axis runs from -1 to 9 and the y axis runs from -5 to 5. The graph is of a decreasing curved function which starts slightly to the right of the y axis. There is no y intercept and the x intercept is at the point (e, 0).\" \/><\/span><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572286777\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572286777\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572286777\">Domain: [latex](0,\\infty)[\/latex], Range: [latex](\u2212\\infty ,\\infty)[\/latex], Vertical asymptote at [latex]x=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572286826\" class=\"exercise\">\n<div id=\"fs-id1170572286828\" class=\"textbox\">\n<p id=\"fs-id1170572286831\"><strong>40.\u00a0<\/strong>[latex]f(x)=\\log x-1[\/latex]<\/p>\n<p><span id=\"fs-id1170572286855\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202804\/CNX_Calc_Figure_01_05_218.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 20 and the y axis runs from -10 to 5. The graph is of an increasing curved function which starts slightly to the right of the y axis. There is no y intercept and the x intercept is at the point (10, 0).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572452755\" class=\"exercise\">\n<div id=\"fs-id1170572452757\" class=\"textbox\">\n<p id=\"fs-id1170572452759\"><strong>41.\u00a0<\/strong>[latex]f(x)=\\ln(x+1)[\/latex]<\/p>\n<p><span id=\"fs-id1170572452796\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202808\/CNX_Calc_Figure_01_05_219.jpg\" alt=\"An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of an increasing curved function which starts slightly to the right of the vertical line \u201cx = -1\u201d. There y intercept and the x intercept are both at the origin.\" \/><\/span><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572452810\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572452810\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572452810\">Domain: [latex](-1,\\infty)[\/latex], Range: [latex](\u2212\\infty ,\\infty)[\/latex], Vertical asymptote at [latex]x=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572452859\">For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and\/or product of logarithms.<\/p>\n<div id=\"fs-id1170572452864\" class=\"exercise\">\n<div id=\"fs-id1170572452866\" class=\"textbox\">\n<p id=\"fs-id1170572452868\"><strong>42.\u00a0<\/strong>[latex]\\log x^4y[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572452904\" class=\"exercise\">\n<div id=\"fs-id1170572452906\" class=\"textbox\">\n<p id=\"fs-id1170572452909\"><strong>43.\u00a0<\/strong>[latex]\\log_3 \\frac{9a^3}{b}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572452934\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572452934\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572452934\">[latex]2+3\\log_3 a-\\log_3 b[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572452966\" class=\"exercise\">\n<div id=\"fs-id1170572452968\" class=\"textbox\">\n<p id=\"fs-id1170572452970\"><strong>44.\u00a0<\/strong>[latex]\\ln a\\sqrt[3]{b}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572546157\" class=\"exercise\">\n<div id=\"fs-id1170572546159\" class=\"textbox\">\n<p id=\"fs-id1170572546161\"><strong>45.\u00a0<\/strong>[latex]\\log_5 \\sqrt{125xy^3}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572546187\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572546187\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572546187\">[latex]\\frac{3}{2}+\\frac{1}{2}\\log_5 x+\\frac{3}{2}\\log_5 y[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572546231\" class=\"exercise\">\n<div id=\"fs-id1170572546233\" class=\"textbox\">\n<p id=\"fs-id1170572546235\"><strong>46.\u00a0<\/strong>[latex]\\log_4 \\frac{\\sqrt[3]{xy}}{64}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572546304\" class=\"exercise\">\n<div id=\"fs-id1170572546306\" class=\"textbox\">\n<p id=\"fs-id1170572546308\"><strong>47.\u00a0<\/strong>[latex]\\ln(\\frac{6}{\\sqrt{e^3}})[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572546336\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572546336\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572546336\">[latex]-\\frac{3}{2}+\\ln 6[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572546356\">For the following exercises, solve the exponential equation exactly.<\/p>\n<div id=\"fs-id1170572546359\" class=\"exercise\">\n<div id=\"fs-id1170572546361\" class=\"textbox\">\n<p id=\"fs-id1170572546363\"><strong>48.\u00a0<\/strong>[latex]5^x=125[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572479927\" class=\"exercise\">\n<div id=\"fs-id1170572479929\" class=\"textbox\">\n<p id=\"fs-id1170572479931\"><strong>49.\u00a0<\/strong>[latex]e^{3x}-15=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572479956\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572479956\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572479956\">[latex]\\frac{\\ln 15}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572479971\" class=\"exercise\">\n<div id=\"fs-id1170572479973\" class=\"textbox\">\n<p id=\"fs-id1170572479975\"><strong>50.\u00a0<\/strong>[latex]8^x=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572480004\" class=\"exercise\">\n<div id=\"fs-id1170572480006\" class=\"textbox\">\n<p id=\"fs-id1170572480008\"><strong>51.\u00a0<\/strong>[latex]4^{x+1}-32=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572480035\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572480035\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572480035\">[latex]\\frac{3}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572480046\" class=\"exercise\">\n<div id=\"fs-id1170572480049\" class=\"textbox\">\n<p id=\"fs-id1170572480051\"><strong>52.\u00a0<\/strong>[latex]3^{x\/14}=\\frac{1}{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572480100\" class=\"exercise\">\n<div id=\"fs-id1170572480102\" class=\"textbox\">\n<p id=\"fs-id1170572480104\"><strong>53.\u00a0<\/strong>[latex]10^x=7.21[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572480122\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572480122\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572480122\">[latex]\\log 7.21[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572480133\" class=\"exercise\">\n<div id=\"fs-id1170572480135\" class=\"textbox\">\n<p id=\"fs-id1170572480137\"><strong>54.\u00a0<\/strong>[latex]4\u00b72^{3x}-20=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572480186\" class=\"exercise\">\n<div id=\"fs-id1170572543576\" class=\"textbox\">\n<p id=\"fs-id1170572543578\"><strong>55.\u00a0<\/strong>[latex]7^{3x-2}=11[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572543602\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572543602\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572543602\">[latex]\\frac{2}{3}+\\frac{\\log 11}{3\\log 7}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572543630\">For the following exercises, solve the logarithmic equation exactly, if possible.<\/p>\n<div id=\"fs-id1170572543633\" class=\"exercise\">\n<div id=\"fs-id1170572543636\" class=\"textbox\">\n<p id=\"fs-id1170572543638\"><strong>56.\u00a0<\/strong>[latex]\\log_3 x=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572543671\" class=\"exercise\">\n<div id=\"fs-id1170572543673\" class=\"textbox\">\n<p id=\"fs-id1170572543675\"><strong>57.\u00a0<\/strong>[latex]\\log_5 x=-2[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572543696\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572543696\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572543696\">[latex]x=\\frac{1}{25}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572543713\" class=\"exercise\">\n<div id=\"fs-id1170572543715\" class=\"textbox\">\n<p id=\"fs-id1170572543717\"><strong>58.\u00a0<\/strong>[latex]\\log_4 (x+5)=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572543761\" class=\"exercise\">\n<div id=\"fs-id1170572543763\" class=\"textbox\">\n<p id=\"fs-id1170572543765\"><strong>59.\u00a0<\/strong>[latex]\\log(2x-7)=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572543794\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572543794\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572543794\">[latex]x=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572543807\" class=\"exercise\">\n<div id=\"fs-id1170572543809\" class=\"textbox\">\n<p id=\"fs-id1170572543811\"><strong>60.\u00a0<\/strong>[latex]\\ln\\sqrt{x+3}=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572088346\" class=\"exercise\">\n<div id=\"fs-id1170572088348\" class=\"textbox\">\n<p id=\"fs-id1170572088350\"><strong>61.\u00a0<\/strong>[latex]\\log_6 (x+9)+\\log_6 x=2[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572088391\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572088391\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572088391\">[latex]x=3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572088404\" class=\"exercise\">\n<div id=\"fs-id1170572088406\" class=\"textbox\">\n<p id=\"fs-id1170572088408\"><strong>62.\u00a0<\/strong>[latex]\\log_4(x+2)-\\log_4(x-1)=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572088466\" class=\"exercise\">\n<div id=\"fs-id1170572088468\" class=\"textbox\">\n<p id=\"fs-id1170572088470\"><strong>63.\u00a0<\/strong>[latex]\\ln x+\\ln (x-2)=\\ln 4[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571120347\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571120347\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571120347\">[latex]1+\\sqrt{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572088504\">For the following exercises, use the change-of-base formula and either base 10 or base [latex]e[\/latex] to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.<\/p>\n<div id=\"fs-id1170572088513\" class=\"exercise\">\n<div id=\"fs-id1170572088515\" class=\"textbox\">\n<p id=\"fs-id1170572088517\"><strong>64.\u00a0<\/strong>[latex]\\log_5 47[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572088562\" class=\"exercise\">\n<div id=\"fs-id1170572088564\" class=\"textbox\">\n<p id=\"fs-id1170572088566\"><strong>65.\u00a0<\/strong>[latex]\\log_7 82[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572088582\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572088582\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572088582\">[latex]\\frac{\\ln 82}{\\ln 7} \\approx 2.2646[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572227257\" class=\"exercise\">\n<div id=\"fs-id1170572227259\" class=\"textbox\">\n<p id=\"fs-id1170572227261\"><strong>66.\u00a0<\/strong>[latex]\\log_6 103[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572227306\" class=\"exercise\">\n<div id=\"fs-id1170572227308\" class=\"textbox\">\n<p id=\"fs-id1170572227310\"><strong>67.\u00a0<\/strong>[latex]\\log_{0.5} 211[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572227327\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572227327\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572227327\">[latex]\\frac{\\ln 211}{\\ln 0.5} \\approx -7.7211[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572227358\" class=\"exercise\">\n<div id=\"fs-id1170572227360\" class=\"textbox\">\n<p id=\"fs-id1170572227362\"><strong>68.\u00a0<\/strong>[latex]{\\text{log}}_{2}\\pi[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572227407\" class=\"exercise\">\n<div id=\"fs-id1170572227409\" class=\"textbox\">\n<p id=\"fs-id1170572227411\"><strong>69.\u00a0<\/strong>[latex]\\log_{0.2} 0.452[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572227428\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572227428\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572227428\">[latex]\\frac{\\ln 0.452}{\\ln 0.2} \\approx 0.4934[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572227457\" class=\"exercise\">\n<div id=\"fs-id1170572227459\" class=\"textbox\">\n<p id=\"fs-id1170572227461\"><strong>70.\u00a0<\/strong>Rewrite the following expressions in terms of exponentials and simplify.<\/p>\n<p id=\"fs-id1170572227465\">a. [latex]2\\cosh(\\ln x)[\/latex]<\/p>\n<p>b. [latex]\\cosh 4x+\\sinh 4x[\/latex]<\/p>\n<p>c. [latex]\\cosh 2x-\\sinh 2x[\/latex]<\/p>\n<p>d. [latex]\\ln(\\cosh x+\\sinh x)+\\ln(\\cosh x-\\sinh x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572230541\" class=\"exercise\">\n<div id=\"fs-id1170572230543\" class=\"textbox\">\n<p id=\"fs-id1170572230545\"><strong>71. [T]<\/strong> The number of bacteria [latex]N[\/latex]\u00a0in a culture after [latex]t[\/latex] days can be modeled by the function [latex]N(t)=1300\u00b72^{t\/4}[\/latex]. Find the number of bacteria present after 15 days.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572230605\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572230605\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572230605\">[latex]\\approx 17,491[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572230620\" class=\"exercise\">\n<div id=\"fs-id1170572230622\" class=\"textbox\">\n<p id=\"fs-id1170572230624\"><strong>72. [T]<\/strong> The demand [latex]D[\/latex] (in millions of barrels) for oil in an oil-rich country is given by the function [latex]D(p)=150\u00b72.7^{-0.25p}[\/latex], where [latex]p[\/latex] is the price (in dollars) of a barrel of oil. Find the amount of oil demanded (to the nearest million barrels) when the price is between $15 and $20.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572230690\" class=\"exercise\">\n<div id=\"fs-id1170572230692\" class=\"textbox\">\n<p id=\"fs-id1170572230695\"><strong>73. [T]<\/strong> The accumulated amount [latex]A[\/latex] of a $100,000 investment whose interest compounds continuously for [latex]t[\/latex] years is given by [latex]A(t)=100,000\u00b7e^{0.055t}[\/latex]. Find the amount [latex]A[\/latex] accumulated in 5 years.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572472846\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572472846\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572472846\">Approximately $131,653 is accumulated in 5 years.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572472851\" class=\"exercise\">\n<div id=\"fs-id1170572472853\" class=\"textbox\">\n<p id=\"fs-id1170572472855\"><strong>74. [T]<\/strong> An investment is compounded monthly, quarterly, or yearly and is given by the function [latex]A=P(1+\\frac{j}{n})^{nt}[\/latex], where [latex]A[\/latex] is the value of the investment at time [latex]t, \\, P[\/latex] is the initial principle that was invested, [latex]j[\/latex] is the annual interest rate, and [latex]n[\/latex] is the number of time the interest is compounded per year. Given a yearly interest rate of 3.5% and an initial principle of $100,000, find the amount [latex]A[\/latex] accumulated in 5 years for interest that is compounded a. daily, b., monthly, c. quarterly, and d. yearly.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572472938\" class=\"exercise\">\n<div id=\"fs-id1170572472940\" class=\"textbox\">\n<p id=\"fs-id1170572472942\"><strong>75. [T]<\/strong> The concentration of hydrogen ions in a substance is denoted by [latex][\\text{H}^{+}][\/latex], measured in moles per liter. The pH of a substance is defined by the logarithmic function [latex]\\text{pH}=\u2212\\log[\\text{H}^{+}][\/latex]. This function is used to measure the acidity of a substance. The pH of water is 7. A substance with a pH less than 7 is an acid, whereas one that has a pH of more than 7 is a base.<\/p>\n<ol id=\"fs-id1170572472994\" style=\"list-style-type: lower-alpha\">\n<li>Find the pH of the following substances. Round answers to one digit.<\/li>\n<li>Determine whether the substance is an acid or a base.\n<ol id=\"fs-id1170572473006\">\n<li>Eggs: [latex][\\text{H}^{+}]=1.6 \\times 10^{-8}[\/latex] mol\/L<\/li>\n<li>Beer: [latex][\\text{H}^{+}]=3.16 \\times 10^{-3}[\/latex] mol\/L<\/li>\n<li>Tomato Juice: [latex][\\text{H}^{+}]=7.94 \\times 10^{-5}[\/latex] mol\/L<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572296436\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572296436\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572296436\">i. a. pH = 8 b. Base ii. a. pH = 3 b. Acid iii. a. pH = 4 b. Acid<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572296441\" class=\"exercise\">\n<div id=\"fs-id1170572296443\" class=\"textbox\">\n<p id=\"fs-id1170572296445\"><strong>76. [T]<\/strong> Iodine-131 is a radioactive substance that decays according to the function [latex]Q(t)=Q_0\u00b7e^{-0.08664t}[\/latex], where [latex]Q_0[\/latex] is the initial quantity of a sample of the substance and [latex]t[\/latex] is in days. Determine how long it takes (to the nearest day) for 95% of a quantity to decay.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572296509\" class=\"exercise\">\n<div id=\"fs-id1170572296511\" class=\"textbox\">\n<p id=\"fs-id1170572296513\"><strong>77. [T]<\/strong> According to the World Bank, at the end of 2013 ([latex]t=0[\/latex]) the U.S. population was 316 million and was increasing according to the following model:<\/p>\n<p id=\"fs-id1170572296531\">[latex]P(t)=316e^{0.0074t}[\/latex],<\/p>\n<p id=\"fs-id1170572296560\">where [latex]P[\/latex] is measured in millions of people and [latex]t[\/latex] is measured in years after 2013.<\/p>\n<ol id=\"fs-id1170572296574\" style=\"list-style-type: lower-alpha\">\n<li>Based on this model, what will be the population of the United States in 2020?<\/li>\n<li>Determine when the U.S. population will be twice what it is in 2013.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572296591\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572296591\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572296591\">a. [latex]\\approx 333[\/latex] million b. 94 years from 2013, or in 2107<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572296604\" class=\"exercise\">\n<div id=\"fs-id1170572296606\" class=\"textbox\">\n<p id=\"fs-id1170572296608\"><strong>78. [T]<\/strong> The amount [latex]A[\/latex] accumulated after 1000 dollars is invested for [latex]t[\/latex] years at an interest rate of 4% is modeled by the function [latex]A(t)=1000(1.04)^t[\/latex].<\/p>\n<ol id=\"fs-id1170572296659\" style=\"list-style-type: lower-alpha\">\n<li>Find the amount accumulated after 5 years and 10 years.<\/li>\n<li>Determine how long it takes for the original investment to triple.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572296689\" class=\"exercise\">\n<div id=\"fs-id1170572296691\" class=\"textbox\">\n<p id=\"fs-id1170572296693\"><strong>79. [T]<\/strong> A bacterial colony grown in a lab is known to double in number in 12 hours. Suppose, initially, there are 1000 bacteria present.<\/p>\n<ol id=\"fs-id1170572212877\" style=\"list-style-type: lower-alpha\">\n<li>Use the exponential function [latex]Q=Q_0e^{kt}[\/latex] to determine the value [latex]k[\/latex], which is the growth rate of the bacteria. Round to four decimal places.<\/li>\n<li>Determine approximately how long it takes for 200,000 bacteria to grow.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572212922\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572212922\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572212922\">a. [latex]k \\approx 0.0578[\/latex] b. [latex]\\approx 92[\/latex] hours<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572212944\" class=\"exercise\">\n<div id=\"fs-id1170572212946\" class=\"textbox\">\n<p id=\"fs-id1170572212949\"><strong>80. [T]<\/strong> The rabbit population on a game reserve doubles every 6 months. Suppose there were 120 rabbits initially.<\/p>\n<ol id=\"fs-id1170572212957\" style=\"list-style-type: lower-alpha\">\n<li>Use the exponential function [latex]P=P_0 a^t[\/latex] to determine the growth rate constant [latex]a[\/latex]. Round to four decimal places.<\/li>\n<li>Use the function in part a. to determine approximately how long it takes for the rabbit population to reach 3500.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572213021\" class=\"exercise\">\n<div id=\"fs-id1170572213023\" class=\"textbox\">\n<p id=\"fs-id1170572213025\"><strong>81. [T]<\/strong> The 1906 earthquake in San Francisco had a magnitude of 8.3 on the Richter scale. At the same time, in Japan, an earthquake with magnitude 4.9 caused only minor damage. Approximately how much more energy was released by the San Francisco earthquake than by the Japanese earthquake?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572213038\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572213038\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572213038\">The San Francisco earthquake had [latex]10^{3.4}[\/latex] or [latex]\\approx 2512[\/latex] times more energy than the Japan earthquake.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572213065\" class=\"review-exercises\"><\/div>\n<div class=\"textbox shaded\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170572544592\" class=\"definition\">\n<dt>base<\/dt>\n<dd id=\"fs-id1170572544597\">the number [latex]b[\/latex] in the exponential function [latex]f(x)=b^x[\/latex] and the logarithmic function [latex]f(x)=\\log_b x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572544649\" class=\"definition\">\n<dt>exponent<\/dt>\n<dd id=\"fs-id1170572544654\">the value [latex]x[\/latex] in the expression [latex]b^x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572544670\" class=\"definition\">\n<dt>hyperbolic functions<\/dt>\n<dd id=\"fs-id1170572544676\">the functions denoted [latex]\\sinh, \\, \\cosh, \\, \\tanh, \\, \\text{csch}, \\, \\text{sech}[\/latex], and [latex]\\coth[\/latex], which involve certain combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572294452\" class=\"definition\">\n<dt>inverse hyperbolic functions<\/dt>\n<dd id=\"fs-id1170572294458\">the inverses of the hyperbolic functions where [latex]\\cosh[\/latex] and [latex]\\text{sech}[\/latex] are restricted to the domain [latex][0,\\infty)[\/latex]; each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572294493\" class=\"definition\">\n<dt>natural exponential function<\/dt>\n<dd id=\"fs-id1170572294499\">the function [latex]f(x)=e^x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572294522\" class=\"definition\">\n<dt>natural logarithm<\/dt>\n<dd id=\"fs-id1170572294527\">the function [latex]\\ln x=\\log_e x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572294549\" class=\"definition\">\n<dt>number e<\/dt>\n<dd id=\"fs-id1170572294554\">as [latex]m[\/latex] gets larger, the quantity [latex](1+(1\/m))^m[\/latex] gets closer to some real number; we define that real number to be [latex]e[\/latex]; the value of [latex]e[\/latex] is approximately 2.718282<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":311,"menu_order":6,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1588","chapter","type-chapter","status-publish","hentry"],"part":1448,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1588","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":15,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1588\/revisions"}],"predecessor-version":[{"id":2581,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1588\/revisions\/2581"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/parts\/1448"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1588\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/media?parent=1588"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapter-type?post=1588"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/contributor?post=1588"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/license?post=1588"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}