{"id":164,"date":"2021-02-03T22:17:38","date_gmt":"2021-02-03T22:17:38","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=164"},"modified":"2022-03-11T21:45:43","modified_gmt":"2022-03-11T21:45:43","slug":"exponential-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/exponential-functions\/","title":{"raw":"Exponential Functions","rendered":"Exponential Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/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<\/ul>\r\n<\/div>\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\" style=\"text-align: center;\">[latex]P(1)=P_0+0.02P_0=P_0(1+0.02)=P_0(1.02)[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572092410\">Its population after 2 years is<\/p>\r\n\r\n<div id=\"fs-id1170572177937\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]P(2)=P(1)+0.02P(1)=P(1)(1.02)=P_0(1.02)^2[\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]P(t)=P_0(1.02)^t[\/latex],<\/div>\r\n&nbsp;\r\n<div><\/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 the table below, 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&nbsp;\r\n<p id=\"fs-id1170572247756\">In Figure 1, 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[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\" \/> Figure 1. 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]\r\n<div id=\"fs-id1170572135350\" class=\"bc-section section\">\r\n<h3>Evaluating Exponential Functions<\/h3>\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 the table below, 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&nbsp;\r\n<div id=\"fs-id1170572169644\" class=\"textbook exercises\">\r\n<h3>Example: Bacterial Growth<\/h3>\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[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 id=\"fs-id1170572173708\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\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[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 id=\"fs-id1170572173826\" class=\"textbox tryit\">\r\n<h3>Interactive<\/h3>\r\n<p id=\"fs-id1170572173829\"><a href=\"https:\/\/www.worldpopulationbalance.org\/understanding-exponential-growth\" target=\"_blank\" rel=\"noopener\">Go to World Population Balance for another example of exponential population growth.<\/a><\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]217504[\/ohm_question]\r\n\r\n<\/div>\r\n<h3>Graphing Exponential Functions<\/h3>\r\nIt may be helpful to recall arrow and interval notation before you explore this section.\r\n<div class=\"textbox examples\">\r\n<h3>Recall: Arrow and interval notation<\/h3>\r\n<table style=\"height: 240px;\">\r\n<thead>\r\n<tr style=\"height: 15px;\">\r\n<th style=\"text-align: center; height: 15px; width: 554.097px;\" colspan=\"2\">Arrow Notation<\/th>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<th style=\"text-align: center; height: 15px; width: 161.875px;\">Symbol<\/th>\r\n<th style=\"text-align: center; height: 15px; width: 380.764px;\">Meaning<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 161.875px;\">[latex]x\\to \\infty[\/latex]<\/td>\r\n<td style=\"height: 30px; width: 380.764px;\">[latex]x[\/latex] approaches infinity ([latex]x[\/latex]\u00a0increases without bound)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 161.875px;\">[latex]x\\to -\\infty [\/latex]<\/td>\r\n<td style=\"height: 30px; width: 380.764px;\">[latex]x[\/latex] approaches negative infinity ([latex]x[\/latex]\u00a0decreases without bound)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 161.875px;\">[latex]f\\left(x\\right)\\to \\infty [\/latex]<\/td>\r\n<td style=\"height: 30px; width: 380.764px;\">the output approaches infinity (the output increases without bound)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 161.875px;\">[latex]f\\left(x\\right)\\to -\\infty [\/latex]<\/td>\r\n<td style=\"height: 30px; width: 380.764px;\">the output approaches negative infinity (the output decreases without bound)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 161.875px;\">[latex]f\\left(x\\right)\\to a[\/latex]<\/td>\r\n<td style=\"height: 30px; width: 380.764px;\">the output approaches [latex]a[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table style=\"height: 312px;\">\r\n<thead>\r\n<tr style=\"height: 30px;\">\r\n<th style=\"height: 30px; width: 194px;\"><\/th>\r\n<th style=\"height: 30px; width: 148px;\">Inequality Notation<\/th>\r\n<th style=\"height: 30px; width: 118px;\">Set-builder Notation<\/th>\r\n<th style=\"height: 30px; width: 80px;\">Interval Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 45px;\">\r\n<td style=\"height: 45px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/1.png\"><img class=\"size-full wp-image-12492 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193534\/1.png\" alt=\"1\" width=\"265\" height=\"60\" \/><\/a><\/td>\r\n<td style=\"height: 45px; width: 148px;\">[latex]5&lt;h\\le10[\/latex]<\/td>\r\n<td style=\"height: 45px; width: 118px;\">[latex]\\{h | 5 &lt; h \\le 10\\}[\/latex]<\/td>\r\n<td style=\"height: 45px; width: 80px;\">[latex](5,10][\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 48px;\">\r\n<td style=\"height: 48px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/2.png\"><img class=\"size-full wp-image-12493 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193535\/2.png\" alt=\"2\" width=\"281\" height=\"75\" \/><\/a><\/td>\r\n<td style=\"height: 48px; width: 148px;\">[latex]5\\le h&lt;10[\/latex]<\/td>\r\n<td style=\"height: 48px; width: 118px;\">[latex]\\{h | 5 \\le h &lt; 10\\}[\/latex]<\/td>\r\n<td style=\"height: 48px; width: 80px;\">[latex][5,10)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 49px;\">\r\n<td style=\"height: 49px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/3.png\"><img class=\"size-full wp-image-12494 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193537\/3.png\" alt=\"3\" width=\"283\" height=\"76\" \/><\/a><\/td>\r\n<td style=\"height: 49px; width: 148px;\">[latex]5&lt;h&lt;10[\/latex]<\/td>\r\n<td style=\"height: 49px; width: 118px;\">[latex]\\{h | 5 &lt; h &lt; 10\\}[\/latex]<\/td>\r\n<td style=\"height: 49px; width: 80px;\">[latex](5,10)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 51px;\">\r\n<td style=\"height: 51px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/4.png\"><img class=\"size-full wp-image-12495 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193538\/4.png\" alt=\"4\" width=\"271\" height=\"76\" \/><\/a><\/td>\r\n<td style=\"height: 51px; width: 148px;\">[latex]h&lt;10[\/latex]<\/td>\r\n<td style=\"height: 51px; width: 118px;\">[latex]\\{h | h &lt; 10\\}[\/latex]<\/td>\r\n<td style=\"height: 51px; width: 80px;\">[latex](-\\infty,10)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 44px;\">\r\n<td style=\"height: 44px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/5.png\"><img class=\"size-full wp-image-12496 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193540\/5.png\" alt=\"5\" width=\"310\" height=\"66\" \/><\/a><\/td>\r\n<td style=\"height: 44px; width: 148px;\">[latex]h&gt;10[\/latex]<\/td>\r\n<td style=\"height: 44px; width: 118px;\">[latex]\\{h | h &gt; 10\\}[\/latex]<\/td>\r\n<td style=\"height: 44px; width: 80px;\">[latex](10,\\infty)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 45px;\">\r\n<td style=\"height: 45px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/6.png\"><img class=\"size-full wp-image-12497 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193542\/6.png\" alt=\"6\" width=\"359\" height=\"67\" \/><\/a><\/td>\r\n<td style=\"height: 45px; width: 148px;\">All real numbers<\/td>\r\n<td style=\"height: 45px; width: 118px;\">[latex]\\mathbf{R}[\/latex]<\/td>\r\n<td style=\"height: 45px; width: 80px;\">[latex](\u2212\\infty,\\infty)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\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] (Figure 2).<\/p>\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\/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\" \/> Figure 2. 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]\r\n<div id=\"fs-id1170572481215\" class=\"textbox tryit\">\r\n<h3>Interactive<\/h3>\r\n<p id=\"fs-id1170572481218\"><a href=\"https:\/\/demonstrations.wolfram.com\/GraphsOfExponentialFunctions\/\" target=\"_blank\" rel=\"noopener\">Visit this site for more exploration of the graphs of exponential functions.<\/a><\/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 class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Laws of Exponents<\/h3>\r\n\r\n<hr \/>\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]\\dfrac{a^x}{b^x} =\\left(\\dfrac{a}{b}\\right)^x[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1170572440093\" class=\"textbook exercises\">\r\n<h3>Example: Using the Laws of Exponents<\/h3>\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[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\" style=\"text-align: center;\">[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\" style=\"text-align: center;\">[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\r\n[caption]Watch the following video to see the worked solution to Example: Using the Laws of Exponents[\/caption]\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tOkk_pSFpzk?controls=0&amp;start=212&amp;end=380&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266833\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/ExponentialAndLogarithmicFunctions212to380_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"1.5 Exponential and Logarithmic Functions\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170572452234\">Use the laws of exponents to simplify [latex]\\dfrac{(6x^{-3}y^2)}{(12x^{-4}y^5)}[\/latex].<\/p>\r\n[reveal-answer q=\"833456\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"833456\"]\r\n<p id=\"fs-id1165042707513\">[latex]x^a\/x^b=x^{a-b}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572452533\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572452533\"]\r\n\r\n[latex]\\dfrac{x}{(2y^3)}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]217501[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>The <strong>Number<\/strong><em><strong>\u00a0[latex]e[\/latex]<\/strong><\/em><\/h2>\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\" style=\"text-align: center;\">[latex]A(1)=P+rP=P(1+r)[\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]A(2)=A(1)+rA(1)=P(1+r)+rP(1+r)=P(1+r)^2[\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]A(t)=P(1+r)^t[\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]A(\\frac{1}{2})=P+(\\frac{r}{2})P=P(1+(\\frac{r}{2}))[\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[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&nbsp;\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\" style=\"text-align: center;\">[latex]A(t)=P\\left(1+\\frac{r}{2}\\right)^{2t}[\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]A(t)=P\\left(1+\\frac{r}{n}\\right)^{nt}[\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]\\left(1+\\frac{r}{n}\\right)^{nt}=\\left(1+\\frac{1}{m}\\right)^{mrt}[\/latex],<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572451344\">and examine the behavior of [latex](1+\\frac{1}{m})^m[\/latex] as [latex]m\\to \\infty[\/latex], using a table of values.<\/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\n&nbsp;\r\n\r\nLooking at this table, it appears that [latex](1+\\frac{1}{m})^m[\/latex] is approaching a number between 2.7 and 2.8 as [latex]m\\to \\infty [\/latex]. In fact, [latex](1+\\frac{1}{m})^m[\/latex] does approach some number as [latex]m\\to \\infty [\/latex]. We call this <strong>number [latex]e[\/latex]<\/strong>. To six decimal places of accuracy,\r\n<div id=\"fs-id1170572549050\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]e \\approx 2.718282[\/latex]<\/div>\r\n&nbsp;\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 Figure 3, 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 module; 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[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\" \/> Figure 3. The graph of [latex]f(x)=e^x[\/latex] has a tangent line with slope 1 at [latex]x=0[\/latex].[\/caption]\r\n<div id=\"fs-id1170572547818\" class=\"textbook exercises\">\r\n<h3>Example: Compounding Interest<\/h3>\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[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\r\n<div style=\"text-align: center;\">[latex]P=\\$500[\/latex] and [latex]r=0.055[\/latex]. Therefore, [latex]A(t)=500e^{0.055t}[\/latex]<\/div><\/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\" style=\"text-align: center;\">[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\" style=\"text-align: center;\">[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\r\n[caption]Watch the following video to see the worked solution to Example: Compounding Interest[\/caption]\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tOkk_pSFpzk?controls=0&amp;start=391&amp;end=555&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266834\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/ExponentialAndLogarithmicFunctions391to555_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"1.5 Exponential and Logarithmic Functions\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1170572455119\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\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[reveal-answer q=\"308667\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"308667\"]\r\n\r\n[latex]A(t)=Pe^{rt}[\/latex]\r\n\r\n[\/hidden-answer]\r\n<p id=\"fs-id1170572455190\">[reveal-answer q=\"505690\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"505690\"]<\/p>\r\n[latex]A(t)=750e^{0.04t}[\/latex]\r\n\r\nAfter 30 years, there will be approximately [latex]\\$2,490.09[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]217509[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/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<\/ul>\n<\/div>\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\" style=\"text-align: center;\">[latex]P(1)=P_0+0.02P_0=P_0(1+0.02)=P_0(1.02)[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572092410\">Its population after 2 years is<\/p>\n<div id=\"fs-id1170572177937\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]P(2)=P(1)+0.02P(1)=P(1)(1.02)=P_0(1.02)^2[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572130048\">In general, its population after [latex]t[\/latex] years is<\/p>\n<div id=\"fs-id1170572280288\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]P(t)=P_0(1.02)^t[\/latex],<\/div>\n<p>&nbsp;<\/p>\n<div><\/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 the table below, 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>&nbsp;<\/p>\n<p id=\"fs-id1170572247756\">In Figure 1, we graph both [latex]y=x^2[\/latex] and [latex]y=2^x[\/latex] to show how the graphs differ.<\/p>\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\">Figure 1. 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 id=\"fs-id1170572135350\" class=\"bc-section section\">\n<h3>Evaluating Exponential Functions<\/h3>\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 the table below, 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<p>&nbsp;<\/p>\n<div id=\"fs-id1170572169644\" class=\"textbook exercises\">\n<h3>Example: Bacterial Growth<\/h3>\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 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 id=\"fs-id1170572173708\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\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 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 id=\"fs-id1170572173826\" class=\"textbox tryit\">\n<h3>Interactive<\/h3>\n<p id=\"fs-id1170572173829\"><a href=\"https:\/\/www.worldpopulationbalance.org\/understanding-exponential-growth\" target=\"_blank\" rel=\"noopener\">Go to World Population Balance for another example of exponential population growth.<\/a><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm217504\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=217504&theme=oea&iframe_resize_id=ohm217504&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h3>Graphing Exponential Functions<\/h3>\n<p>It may be helpful to recall arrow and interval notation before you explore this section.<\/p>\n<div class=\"textbox examples\">\n<h3>Recall: Arrow and interval notation<\/h3>\n<table style=\"height: 240px;\">\n<thead>\n<tr style=\"height: 15px;\">\n<th style=\"text-align: center; height: 15px; width: 554.097px;\" colspan=\"2\">Arrow Notation<\/th>\n<\/tr>\n<tr style=\"height: 15px;\">\n<th style=\"text-align: center; height: 15px; width: 161.875px;\">Symbol<\/th>\n<th style=\"text-align: center; height: 15px; width: 380.764px;\">Meaning<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 161.875px;\">[latex]x\\to \\infty[\/latex]<\/td>\n<td style=\"height: 30px; width: 380.764px;\">[latex]x[\/latex] approaches infinity ([latex]x[\/latex]\u00a0increases without bound)<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 161.875px;\">[latex]x\\to -\\infty[\/latex]<\/td>\n<td style=\"height: 30px; width: 380.764px;\">[latex]x[\/latex] approaches negative infinity ([latex]x[\/latex]\u00a0decreases without bound)<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 161.875px;\">[latex]f\\left(x\\right)\\to \\infty[\/latex]<\/td>\n<td style=\"height: 30px; width: 380.764px;\">the output approaches infinity (the output increases without bound)<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 161.875px;\">[latex]f\\left(x\\right)\\to -\\infty[\/latex]<\/td>\n<td style=\"height: 30px; width: 380.764px;\">the output approaches negative infinity (the output decreases without bound)<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 161.875px;\">[latex]f\\left(x\\right)\\to a[\/latex]<\/td>\n<td style=\"height: 30px; width: 380.764px;\">the output approaches [latex]a[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"height: 312px;\">\n<thead>\n<tr style=\"height: 30px;\">\n<th style=\"height: 30px; width: 194px;\"><\/th>\n<th style=\"height: 30px; width: 148px;\">Inequality Notation<\/th>\n<th style=\"height: 30px; width: 118px;\">Set-builder Notation<\/th>\n<th style=\"height: 30px; width: 80px;\">Interval Notation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 45px;\">\n<td style=\"height: 45px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12492 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193534\/1.png\" alt=\"1\" width=\"265\" height=\"60\" \/><\/a><\/td>\n<td style=\"height: 45px; width: 148px;\">[latex]5<h\\le10[\/latex]<\/td>\n<td style=\"height: 45px; width: 118px;\">[latex]\\{h | 5 < h \\le 10\\}[\/latex]<\/td>\n<td style=\"height: 45px; width: 80px;\">[latex](5,10][\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 48px;\">\n<td style=\"height: 48px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12493 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193535\/2.png\" alt=\"2\" width=\"281\" height=\"75\" \/><\/a><\/td>\n<td style=\"height: 48px; width: 148px;\">[latex]5\\le h<10[\/latex]<\/td>\n<td style=\"height: 48px; width: 118px;\">[latex]\\{h | 5 \\le h < 10\\}[\/latex]<\/td>\n<td style=\"height: 48px; width: 80px;\">[latex][5,10)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 49px;\">\n<td style=\"height: 49px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/3.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12494 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193537\/3.png\" alt=\"3\" width=\"283\" height=\"76\" \/><\/a><\/td>\n<td style=\"height: 49px; width: 148px;\">[latex]5<h<10[\/latex]<\/td>\n<td style=\"height: 49px; width: 118px;\">[latex]\\{h | 5 < h < 10\\}[\/latex]<\/td>\n<td style=\"height: 49px; width: 80px;\">[latex](5,10)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 51px;\">\n<td style=\"height: 51px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/4.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12495 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193538\/4.png\" alt=\"4\" width=\"271\" height=\"76\" \/><\/a><\/td>\n<td style=\"height: 51px; width: 148px;\">[latex]h<10[\/latex]<\/td>\n<td style=\"height: 51px; width: 118px;\">[latex]\\{h | h < 10\\}[\/latex]<\/td>\n<td style=\"height: 51px; width: 80px;\">[latex](-\\infty,10)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td style=\"height: 44px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/5.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12496 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193540\/5.png\" alt=\"5\" width=\"310\" height=\"66\" \/><\/a><\/td>\n<td style=\"height: 44px; width: 148px;\">[latex]h>10[\/latex]<\/td>\n<td style=\"height: 44px; width: 118px;\">[latex]\\{h | h > 10\\}[\/latex]<\/td>\n<td style=\"height: 44px; width: 80px;\">[latex](10,\\infty)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 45px;\">\n<td style=\"height: 45px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/6.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12497 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193542\/6.png\" alt=\"6\" width=\"359\" height=\"67\" \/><\/a><\/td>\n<td style=\"height: 45px; width: 148px;\">All real numbers<\/td>\n<td style=\"height: 45px; width: 118px;\">[latex]\\mathbf{R}[\/latex]<\/td>\n<td style=\"height: 45px; width: 80px;\">[latex](\u2212\\infty,\\infty)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\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] (Figure 2).<\/p>\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\">Figure 2. 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 id=\"fs-id1170572481215\" class=\"textbox tryit\">\n<h3>Interactive<\/h3>\n<p id=\"fs-id1170572481218\"><a href=\"https:\/\/demonstrations.wolfram.com\/GraphsOfExponentialFunctions\/\" target=\"_blank\" rel=\"noopener\">Visit this site for more exploration of the graphs of exponential functions.<\/a><\/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 class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Laws of Exponents<\/h3>\n<hr \/>\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]\\dfrac{a^x}{b^x} =\\left(\\dfrac{a}{b}\\right)^x[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170572440093\" class=\"textbook exercises\">\n<h3>Example: Using the Laws of Exponents<\/h3>\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 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\" style=\"text-align: center;\">[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\" style=\"text-align: center;\">[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<p>Watch the following video to see the worked solution to Example: Using the Laws of Exponents<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tOkk_pSFpzk?controls=0&amp;start=212&amp;end=380&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/ExponentialAndLogarithmicFunctions212to380_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;1.5 Exponential and Logarithmic Functions&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170572452234\">Use the laws of exponents to simplify [latex]\\dfrac{(6x^{-3}y^2)}{(12x^{-4}y^5)}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q833456\">Hint<\/span><\/p>\n<div id=\"q833456\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042707513\">[latex]x^a\/x^b=x^{a-b}[\/latex]<\/p>\n<\/div>\n<\/div>\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\">\n<p>[latex]\\dfrac{x}{(2y^3)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm217501\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=217501&theme=oea&iframe_resize_id=ohm217501&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>The <strong>Number<\/strong><em><strong>\u00a0[latex]e[\/latex]<\/strong><\/em><\/h2>\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\" style=\"text-align: center;\">[latex]A(1)=P+rP=P(1+r)[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572449156\">The amount of money after 2 years is<\/p>\n<div id=\"fs-id1170572449164\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A(2)=A(1)+rA(1)=P(1+r)+rP(1+r)=P(1+r)^2[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572449268\">More generally, the amount after [latex]t[\/latex] years is<\/p>\n<div id=\"fs-id1170572449276\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A(t)=P(1+r)^t[\/latex]<\/div>\n<p>&nbsp;<\/p>\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\" style=\"text-align: center;\">[latex]A(\\frac{1}{2})=P+(\\frac{r}{2})P=P(1+(\\frac{r}{2}))[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572548410\">The amount of money after 1 year is<\/p>\n<div id=\"fs-id1170572548417\" class=\"equation unnumbered\" style=\"text-align: center;\">[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>&nbsp;<\/p>\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\" style=\"text-align: center;\">[latex]A(t)=P\\left(1+\\frac{r}{2}\\right)^{2t}[\/latex]<\/div>\n<p>&nbsp;<\/p>\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\" style=\"text-align: center;\">[latex]A(t)=P\\left(1+\\frac{r}{n}\\right)^{nt}[\/latex]<\/div>\n<p>&nbsp;<\/p>\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\" style=\"text-align: center;\">[latex]\\left(1+\\frac{r}{n}\\right)^{nt}=\\left(1+\\frac{1}{m}\\right)^{mrt}[\/latex],<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572451344\">and examine the behavior of [latex](1+\\frac{1}{m})^m[\/latex] as [latex]m\\to \\infty[\/latex], using a table of values.<\/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>&nbsp;<\/p>\n<p>Looking at this table, it appears that [latex](1+\\frac{1}{m})^m[\/latex] is approaching a number between 2.7 and 2.8 as [latex]m\\to \\infty[\/latex]. In fact, [latex](1+\\frac{1}{m})^m[\/latex] does approach some number as [latex]m\\to \\infty[\/latex]. We call this <strong>number [latex]e[\/latex]<\/strong>. To six decimal places of accuracy,<\/p>\n<div id=\"fs-id1170572549050\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]e \\approx 2.718282[\/latex]<\/div>\n<p>&nbsp;<\/p>\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 Figure 3, 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 module; 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 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\">Figure 3. 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 id=\"fs-id1170572547818\" class=\"textbook exercises\">\n<h3>Example: Compounding Interest<\/h3>\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 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\n<div style=\"text-align: center;\">[latex]P=\\$500[\/latex] and [latex]r=0.055[\/latex]. Therefore, [latex]A(t)=500e^{0.055t}[\/latex]<\/div>\n<\/li>\n<li>After 10 years, the amount of money in the account is\n<div id=\"fs-id1170573386623\" class=\"equation unnumbered\" style=\"text-align: center;\">[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\" style=\"text-align: center;\">[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<p>Watch the following video to see the worked solution to Example: Compounding Interest<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tOkk_pSFpzk?controls=0&amp;start=391&amp;end=555&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266834\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/ExponentialAndLogarithmicFunctions391to555_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;1.5 Exponential and Logarithmic Functions&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1170572455119\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q308667\">Hint<\/span><\/p>\n<div id=\"q308667\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]A(t)=Pe^{rt}[\/latex]<\/p>\n<\/div>\n<\/div>\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 Solution<\/span><\/p>\n<div id=\"q505690\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]A(t)=750e^{0.04t}[\/latex]<\/p>\n<p>After 30 years, there will be approximately [latex]\\$2,490.09[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm217509\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=217509&theme=oea&iframe_resize_id=ohm217509&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-164\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>1.5 Exponential and Logarithmic Functions. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":22,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"1.5 Exponential and Logarithmic Functions\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-164","chapter","type-chapter","status-publish","hentry"],"part":21,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/164","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":38,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/164\/revisions"}],"predecessor-version":[{"id":4756,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/164\/revisions\/4756"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/21"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/164\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=164"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=164"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=164"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=164"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}