{"id":2196,"date":"2018-01-11T21:33:33","date_gmt":"2018-01-11T21:33:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/integrals-exponential-functions-and-logarithms\/"},"modified":"2018-02-20T16:40:32","modified_gmt":"2018-02-20T16:40:32","slug":"integrals-exponential-functions-and-logarithms","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/chapter\/integrals-exponential-functions-and-logarithms\/","title":{"raw":"6.7 Integrals, Exponential Functions, and Logarithms","rendered":"6.7 Integrals, Exponential Functions, and Logarithms"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Write the definition of the natural logarithm as an integral.<\/li>\r\n \t<li>Recognize the derivative of the natural logarithm.<\/li>\r\n \t<li>Integrate functions involving the natural logarithmic function.<\/li>\r\n \t<li>Define the number [latex]e[\/latex] through an integral.<\/li>\r\n \t<li>Recognize the derivative and integral of the exponential function.<\/li>\r\n \t<li>Prove properties of logarithms and exponential functions using integrals.<\/li>\r\n \t<li>Express general logarithmic and exponential functions in terms of natural logarithms and exponentials.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1167793957648\">We already examined exponential functions and logarithms in earlier chapters. However, we glossed over some key details in the previous discussions. For example, we did not study how to treat exponential functions with exponents that are irrational. The definition of the number [latex]e[\/latex] is another area where the previous development was somewhat incomplete. We now have the tools to deal with these concepts in a more mathematically rigorous way, and we do so in this section.<\/p>\r\n<p id=\"fs-id1167793417231\">For purposes of this section, assume we have not yet defined the natural logarithm, the number [latex]e[\/latex], or any of the integration and differentiation formulas associated with these functions. By the end of the section, we will have studied these concepts in a mathematically rigorous way (and we will see they are consistent with the concepts we learned earlier).<\/p>\r\nWe begin the section by defining the natural logarithm in terms of an integral. This definition forms the foundation for the section. From this definition, we derive differentiation formulas, define the number [latex]e,[\/latex] and expand these concepts to logarithms and exponential functions of any base.\r\n<div id=\"fs-id1167794054166\" class=\"bc-section section\">\r\n<h1>The Natural Logarithm as an Integral<\/h1>\r\n<p id=\"fs-id1167793958127\">Recall the power rule for integrals:<\/p>\r\n\r\n<div id=\"fs-id1167793299462\" class=\"equation unnumbered\">[latex]\\int {x}^{n}dx=\\frac{{x}^{n+1}}{n+1}+C,n\\ne \\text{\u2212}1.[\/latex]<\/div>\r\nClearly, this does not work when [latex]n=-1,[\/latex] as it would force us to divide by zero. So, what do we do with [latex]\\int \\frac{1}{x}dx?[\/latex] Recall from the Fundamental Theorem of Calculus that [latex]{\\int }_{1}^{x}\\frac{1}{t}dt[\/latex] is an antiderivative of [latex]1\\text{\/}x.[\/latex] Therefore, we can make the following definition.\r\n<div id=\"fs-id1167793370244\" class=\"textbox key-takeaways\">\r\n<div class=\"title\">\r\n<h3>Definition<\/h3>\r\n<\/div>\r\n<p id=\"fs-id1167793953542\">For [latex]x&gt;0,[\/latex] define the natural logarithm function by<\/p>\r\n\r\n<div id=\"fs-id1167793550098\" class=\"equation\">[latex]\\text{ln}x={\\int }_{1}^{x}\\frac{1}{t}dt.[\/latex]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793617974\">For [latex]x&gt;1,[\/latex] this is just the area under the curve [latex]y=1\\text{\/}t[\/latex] from 1 to [latex]x.[\/latex] For [latex]x&lt;1,[\/latex] we have [latex]{\\int }_{1}^{x}\\frac{1}{t}dt=\\text{\u2212}{\\int }_{x}^{1}\\frac{1}{t}dt,[\/latex] so in this case it is the negative of the area under the curve from [latex]x\\text{ to }1[\/latex] (see the following figure).<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_06_07_001\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"589\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213320\/CNX_Calc_Figure_06_07_001.jpg\" alt=\"This figure has two graphs. The first is the curve y=1\/t. It is decreasing and in the first quadrant. Under the curve is a shaded area. The area is bounded to the left at x=1. The area is labeled \u201carea=lnx\u201d. The second graph is the same curve y=1\/t. It has shaded area under the curve bounded to the right by x=1. It is labeled \u201carea=-lnx\u201d.\" width=\"589\" height=\"311\" \/> Figure 1. (a) When [latex]x&gt;1,[\/latex] the natural logarithm is the area under the curve [latex]y=1\\text{\/}t[\/latex] from [latex]1\\text{ to }x.[\/latex] (b) When [latex]x&lt;1,[\/latex] the natural logarithm is the negative of the area under the curve from [latex]x[\/latex] to 1.[\/caption]<\/div>\r\n<p id=\"fs-id1167793932688\">Notice that [latex]\\text{ln}1=0.[\/latex] Furthermore, the function [latex]y=1\\text{\/}t&gt;0[\/latex] for [latex]x&gt;0.[\/latex] Therefore, by the properties of integrals, it is clear that [latex]\\text{ln}x[\/latex] is increasing for [latex]x&gt;0.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1167793930384\" class=\"bc-section section\">\r\n<h1>Properties of the Natural Logarithm<\/h1>\r\n<p id=\"fs-id1167794139011\">Because of the way we defined the natural logarithm, the following differentiation formula falls out immediately as a result of to the Fundamental Theorem of Calculus.<\/p>\r\n\r\n<div id=\"fs-id1167794186831\" class=\"textbox key-takeaways theorem\">\r\n<h3>Derivative of the Natural Logarithm<\/h3>\r\n<p id=\"fs-id1167794050615\">For [latex]x&gt;0,[\/latex] the derivative of the natural logarithm is given by<\/p>\r\n\r\n<div id=\"fs-id1167793432137\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}\\text{ln}x=\\frac{1}{x}.[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793879890\" class=\"textbox key-takeaways theorem\">\r\n<h3>Corollary to the Derivative of the Natural Logarithm<\/h3>\r\n<p id=\"fs-id1167794128288\">The function [latex]\\text{ln}x[\/latex] is differentiable; therefore, it is continuous.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1167794141119\">A graph of [latex]\\text{ln}x[\/latex] is shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_06_07_002\">(Figure)<\/a>. Notice that it is continuous throughout its domain of [latex](0,\\infty ).[\/latex]<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_06_07_002\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"266\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213322\/CNX_Calc_Figure_06_07_002.jpg\" alt=\"This figure is a graph. It is an increasing curve labeled f(x)=lnx. The curve is increasing with the y-axis as an asymptote. The curve intersects the x-axis at x=1.\" width=\"266\" height=\"347\" \/> Figure 2. The graph of [latex]f(x)=\\text{ln}x[\/latex] shows that it is a continuous function.[\/caption]<\/div>\r\n<div id=\"fs-id1167793967054\" class=\"textbox examples\">\r\n<h3>Calculating Derivatives of Natural Logarithms<\/h3>\r\n<div id=\"fs-id1167793424342\" class=\"exercise\">\r\n<div id=\"fs-id1167793425720\" class=\"textbox\">\r\n<p id=\"fs-id1167794168088\">Calculate the following derivatives:<\/p>\r\n\r\n<ol id=\"fs-id1167793966989\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dx}\\text{ln}(5{x}^{3}-2)[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}{(\\text{ln}(3x))}^{2}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167794171377\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794171377\"]\r\n<p id=\"fs-id1167794171377\">We need to apply the chain rule in both cases.<\/p>\r\n\r\n<ol id=\"fs-id1167793926297\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dx}\\text{ln}(5{x}^{3}-2)=\\frac{15{x}^{2}}{5{x}^{3}-2}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}{(\\text{ln}(3x))}^{2}=\\frac{2(\\text{ln}(3x))\u00b73}{3x}=\\frac{2(\\text{ln}(3x))}{x}[\/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-id1167793398778\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1167793369601\" class=\"exercise\">\r\n<div id=\"fs-id1167794052465\" class=\"textbox\">\r\n<p id=\"fs-id1167794334473\">Calculate the following derivatives:<\/p>\r\n\r\n<ol id=\"fs-id1167793964608\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dx}\\text{ln}(2{x}^{2}+x)[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}{(\\text{ln}({x}^{3}))}^{2}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793259644\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793259644\"]\r\n<ol id=\"fs-id1167793259644\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dx}\\text{ln}(2{x}^{2}+x)=\\frac{4x+1}{2{x}^{2}+x}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}{(\\text{ln}({x}^{3}))}^{2}=\\frac{6\\text{ln}({x}^{3})}{x}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1167793277086\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1167794187630\">Apply the differentiation formula just provided and use the chain rule as necessary.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793609503\">Note that if we use the absolute value function and create a new function [latex]\\text{ln}|x|,[\/latex] we can extend the domain of the natural logarithm to include [latex]x&lt;0.[\/latex] Then [latex](d\\text{\/}(dx))\\text{ln}|x|=1\\text{\/}x.[\/latex] This gives rise to the familiar integration formula.<\/p>\r\n\r\n<div id=\"fs-id1167793432204\" class=\"textbox key-takeaways theorem\">\r\n<h3>Integral of (1\/[latex]u[\/latex]) <em>du<\/em><\/h3>\r\n<p id=\"fs-id1167793832055\">The natural logarithm is the antiderivative of the function [latex]f(u)=1\\text{\/}u\\text{:}[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1167793973017\" class=\"equation unnumbered\">[latex]\\int \\frac{1}{u}du=\\text{ln}|u|+C.[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793423035\" class=\"textbox examples\">\r\n<h3>Calculating Integrals Involving Natural Logarithms<\/h3>\r\n<div id=\"fs-id1167793948151\" class=\"exercise\">\r\n<div id=\"fs-id1167794027994\" class=\"textbox\">\r\n<p id=\"fs-id1167793989365\">Calculate the integral [latex]\\int \\frac{x}{{x}^{2}+4}dx.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793887223\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793887223\"]\r\n<p id=\"fs-id1167793887223\">Using [latex]u[\/latex]-substitution, let [latex]u={x}^{2}+4.[\/latex] Then [latex]du=2xdx[\/latex] and we have<\/p>\r\n\r\n<div id=\"fs-id1167793971678\" class=\"equation unnumbered\">[latex]\\int \\frac{x}{{x}^{2}+4}dx=\\frac{1}{2}\\int \\frac{1}{u}du\\frac{1}{2}\\text{ln}|u|+C=\\frac{1}{2}\\text{ln}|{x}^{2}+4|+C=\\frac{1}{2}\\text{ln}({x}^{2}+4)+C.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794041523\" class=\"textbox exercises checkpoint\">\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1167793377877\" class=\"textbox\">\r\n<p id=\"fs-id1167793414202\">Calculate the integral [latex]\\int \\frac{{x}^{2}}{{x}^{3}+6}dx.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793525036\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793525036\"]\r\n<p id=\"fs-id1167793525036\">[latex]\\int \\frac{{x}^{2}}{{x}^{3}+6}dx=\\frac{1}{3}\\text{ln}|{x}^{3}+6|+C[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1167793549644\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1167793862633\">Apply the integration formula provided earlier and use [latex]u[\/latex]-substitution as necessary.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167794062403\">Although we have called our function a \u201clogarithm,\u201d we have not actually proved that any of the properties of logarithms hold for this function. We do so here.<\/p>\r\n\r\n<div id=\"fs-id1167793366787\" class=\"textbox key-takeaways theorem\">\r\n<h3>Properties of the Natural Logarithm<\/h3>\r\n<p id=\"fs-id1167793936147\">If [latex]a,b&gt;0[\/latex] and [latex]r[\/latex] is a rational number, then<\/p>\r\n\r\n<ol id=\"fs-id1167793271023\">\r\n \t<li>[latex]\\text{ln}1=0[\/latex]<\/li>\r\n \t<li>[latex]\\text{ln}(ab)=\\text{ln}a+\\text{ln}b[\/latex]<\/li>\r\n \t<li>[latex]\\text{ln}(\\frac{a}{b})=\\text{ln}a-\\text{ln}b[\/latex]<\/li>\r\n \t<li>[latex]\\text{ln}({a}^{r})=r\\text{ln}a[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1167793509941\" class=\"bc-section section\">\r\n<h2>Proof<\/h2>\r\n<ol id=\"fs-id1167793977078\">\r\n \t<li>By definition, [latex]\\text{ln}1={\\int }_{1}^{1}\\frac{1}{t}dt=0.[\/latex]<\/li>\r\n \t<li>We have\r\n<div id=\"fs-id1167793964775\" class=\"equation unnumbered\">[latex]\\text{ln}(ab)={\\int }_{1}^{ab}\\frac{1}{t}dt={\\int }_{1}^{a}\\frac{1}{t}dt+{\\int }_{a}^{ab}\\frac{1}{t}dt.[\/latex]<\/div>\r\nUse [latex]u\\text{-substitution}[\/latex] on the last integral in this expression. Let [latex]u=t\\text{\/}a.[\/latex] Then [latex]du=(1\\text{\/}a)dt.[\/latex] Furthermore, when [latex]t=a,u=1,[\/latex] and when [latex]t=ab,u=b.[\/latex] So we get\r\n<div id=\"fs-id1167793490878\" class=\"equation unnumbered\">[latex]\\text{ln}(ab)={\\int }_{1}^{a}\\frac{1}{t}dt+{\\int }_{a}^{ab}\\frac{1}{t}dt={\\int }_{1}^{a}\\frac{1}{t}dt+{\\int }_{1}^{ab}\\frac{a}{t}\u00b7\\frac{1}{a}dt={\\int }_{1}^{a}\\frac{1}{t}dt+{\\int }_{1}^{b}\\frac{1}{u}du=\\text{ln}a+\\text{ln}b.[\/latex]<\/div><\/li>\r\n \t<li>Note that\r\n<div id=\"fs-id1167793951598\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}\\text{ln}({x}^{r})=\\frac{r{x}^{r-1}}{{x}^{r}}=\\frac{r}{x}.[\/latex]<\/div>\r\nFurthermore,\r\n<div class=\"equation unnumbered\">[latex]\\frac{d}{dx}(r\\text{ln}x)=\\frac{r}{x}.[\/latex]<\/div>\r\nSince the derivatives of these two functions are the same, by the Fundamental Theorem of Calculus, they must differ by a constant. So we have\r\n<div id=\"fs-id1167793563854\" class=\"equation unnumbered\">[latex]\\text{ln}({x}^{r})=r\\text{ln}x+C[\/latex]<\/div>\r\nfor some constant [latex]C.[\/latex] Taking [latex]x=1,[\/latex] we get\r\n<div id=\"fs-id1167793510762\" class=\"equation unnumbered\">[latex]\\begin{array}{ccc}\\hfill \\text{ln}({1}^{r})&amp; =\\hfill &amp; r\\text{ln}(1)+C\\hfill \\\\ \\hfill 0&amp; =\\hfill &amp; r(0)+C\\hfill \\\\ \\hfill C&amp; =\\hfill &amp; 0.\\hfill \\end{array}[\/latex]<\/div>\r\nThus [latex]\\text{ln}({x}^{r})=r\\text{ln}x[\/latex] and the proof is complete. Note that we can extend this property to irrational values of [latex]r[\/latex] later in this section.\r\nPart iii. follows from parts ii. and iv. and the proof is left to you.<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1167794210729\">\u25a1<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<div id=\"fs-id1167794077125\" class=\"exercise\">\r\n<div id=\"fs-id1167793961386\" class=\"textbox\">\r\n<h3>Using Properties of Logarithms<\/h3>\r\n<p id=\"fs-id1167794003628\">Use properties of logarithms to simplify the following expression into a single logarithm:<\/p>\r\n\r\n<div id=\"fs-id1167793590440\" class=\"equation unnumbered\">[latex]\\text{ln}9-2\\text{ln}3+\\text{ln}(\\frac{1}{3}).[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167794337026\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794337026\"]\r\n<p id=\"fs-id1167794337026\">We have<\/p>\r\n\r\n<div id=\"fs-id1167794337029\" class=\"equation unnumbered\">[latex]\\text{ln}9-2\\text{ln}3+\\text{ln}(\\frac{1}{3})=\\text{ln}({3}^{2})-2\\text{ln}3+\\text{ln}({3}^{-1})=2\\text{ln}3-2\\text{ln}3-\\text{ln}3=\\text{\u2212}\\text{ln}3.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793603772\" class=\"textbox exercises checkpoint\">\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1167793294690\" class=\"textbox\">\r\n<p id=\"fs-id1167793607833\">Use properties of logarithms to simplify the following expression into a single logarithm:<\/p>\r\n\r\n<div id=\"fs-id1167793607837\" class=\"equation unnumbered\">[latex]\\text{ln}8-\\text{ln}2-\\text{ln}(\\frac{1}{4}).[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793949542\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793949542\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793949542\"][latex]4\\text{ln}2[\/latex]<\/div>\r\n<div id=\"fs-id1167794144762\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1167793949514\">Apply the properties of logarithms.<\/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-id1167793514534\" class=\"bc-section section\">\r\n<h1>Defining the Number [latex]e[\/latex]<\/h1>\r\n<p id=\"fs-id1167793877994\">Now that we have the natural logarithm defined, we can use that function to define the number [latex]e.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1167793249163\" class=\"textbox key-takeaways\">\r\n<div class=\"title\">\r\n<h3>Definition<\/h3>\r\n<\/div>\r\n<p id=\"fs-id1167793249167\">The number [latex]e[\/latex] is defined to be the real number such that<\/p>\r\n\r\n<div id=\"fs-id1167794140148\" class=\"equation unnumbered\">[latex]\\text{ln}e=1.[\/latex]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793871307\">To put it another way, the area under the curve [latex]y=1\\text{\/}t[\/latex] between [latex]t=1[\/latex] and [latex]t=e[\/latex] is 1 (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_06_07_003\">(Figure)<\/a>). The proof that such a number exists and is unique is left to you. (<em>Hint<\/em>: Use the Intermediate Value Theorem to prove existence and the fact that [latex]\\text{ln}x[\/latex] is increasing to prove uniqueness.)<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_06_07_003\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"304\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213325\/CNX_Calc_Figure_06_07_003.jpg\" alt=\"This figure is a graph. It is the curve y=1\/t. It is decreasing and in the first quadrant. Under the curve is a shaded area. The area is bounded to the left at x=1 and to the right at x=e. The area is labeled \u201carea=1\u201d.\" width=\"304\" height=\"316\" \/> Figure 3. The area under the curve from 1 to [latex]e[\/latex] is equal to one.[\/caption]<\/div>\r\n<p id=\"fs-id1167793939504\">The number [latex]e[\/latex] can be shown to be irrational, although we won\u2019t do so here (see the Student Project in <a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/HTmjSAcf@2.46:CZi7x6Ls@3\/Taylor-and-Maclaurin-Series\">Taylor and Maclaurin Series<\/a> in the second volume of this text). Its approximate value is given by<\/p>\r\n\r\n<div id=\"fs-id1167793443440\" class=\"equation unnumbered\">[latex]e\\approx 2.71828182846.[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794051412\" class=\"bc-section section\">\r\n<h1>The Exponential Function<\/h1>\r\n<p id=\"fs-id1167794051417\">We now turn our attention to the function [latex]{e}^{x}.[\/latex] Note that the natural logarithm is one-to-one and therefore has an inverse function. For now, we denote this inverse function by [latex]\\text{exp}x.[\/latex] Then,<\/p>\r\n\r\n<div id=\"fs-id1167793514717\" class=\"equation unnumbered\">[latex]\\text{exp}(\\text{ln}x)=x\\text{ for }x&gt;0\\text{ and }\\text{ln}(\\text{exp}x)=x\\text{for all}x.[\/latex]<\/div>\r\n<p id=\"fs-id1167793932178\">The following figure shows the graphs of [latex]\\text{exp}x[\/latex] and [latex]\\text{ln}x.[\/latex]<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_06_07_004\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"421\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213328\/CNX_Calc_Figure_06_07_004.jpg\" alt=\"This figure is a graph. It has three curves. The first curve is labeled exp x. It is an increasing curve with the x-axis as a horizontal asymptote. It intersects the y-axis at y=1. The second curve is a diagonal line through the origin. The third curve is labeled lnx. It is an increasing curve with the y-axis as an vertical axis. It intersects the x-axis at x=1.\" width=\"421\" height=\"422\" \/> Figure 4. The graphs of [latex]\\text{ln}x[\/latex] and [latex]\\text{exp}x.[\/latex][\/caption]<\/div>\r\n<p id=\"fs-id1167793905821\">We hypothesize that [latex]\\text{exp}x={e}^{x}.[\/latex] For rational values of [latex]x,[\/latex] this is easy to show. If [latex]x[\/latex] is rational, then we have [latex]\\text{ln}({e}^{x})=x\\text{ln}e=x.[\/latex] Thus, when [latex]x[\/latex] is rational, [latex]{e}^{x}=\\text{exp}x.[\/latex] For irrational values of [latex]x,[\/latex] we simply define [latex]{e}^{x}[\/latex] as the inverse function of [latex]\\text{ln}x.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1167793579578\" class=\"textbox key-takeaways\">\r\n<div class=\"title\">\r\n<h3>Definition<\/h3>\r\n<\/div>\r\n<p id=\"fs-id1167793367959\">For any real number [latex]x,[\/latex] define [latex]y={e}^{x}[\/latex] to be the number for which<\/p>\r\n\r\n<div id=\"fs-id1167793447883\" class=\"equation\">[latex]\\text{ln}y=\\text{ln}({e}^{x})=x.[\/latex]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793489536\">Then we have [latex]{e}^{x}=\\text{exp}(x)[\/latex] for all [latex]x,[\/latex] and thus<\/p>\r\n\r\n<div id=\"fs-id1167793564127\" class=\"equation\">[latex]{e}^{\\text{ln}x}=x\\text{ for }x&gt;0\\text{ and }\\text{ln}({e}^{x})=x[\/latex]<\/div>\r\n<p id=\"fs-id1167794036848\">for all [latex]x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1167793964114\" class=\"bc-section section\">\r\n<h1>Properties of the Exponential Function<\/h1>\r\n<p id=\"fs-id1167793516179\">Since the exponential function was defined in terms of an inverse function, and not in terms of a power of [latex]e,[\/latex] we must verify that the usual laws of exponents hold for the function [latex]{e}^{x}.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1167794176164\" class=\"textbox key-takeaways theorem\">\r\n<h3>Properties of the Exponential Function<\/h3>\r\n<p id=\"fs-id1167794210866\">If [latex]p[\/latex] and [latex]q[\/latex] are any real numbers and [latex]r[\/latex] is a rational number, then<\/p>\r\n\r\n<ol id=\"fs-id1167793361174\">\r\n \t<li>[latex]{e}^{p}{e}^{q}={e}^{p+q}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{{e}^{p}}{{e}^{q}}={e}^{p-q}[\/latex]<\/li>\r\n \t<li>[latex]{({e}^{p})}^{r}={e}^{pr}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1167794064034\" class=\"bc-section section\">\r\n<h2>Proof<\/h2>\r\n<p id=\"fs-id1167793420459\">Note that if [latex]p[\/latex] and [latex]q[\/latex] are rational, the properties hold. However, if [latex]p[\/latex] or [latex]q[\/latex] are irrational, we must apply the inverse function definition of [latex]{e}^{x}[\/latex] and verify the properties. Only the first property is verified here; the other two are left to you. We have<\/p>\r\n\r\n<div id=\"fs-id1167793301084\" class=\"equation unnumbered\">[latex]\\text{ln}({e}^{p}{e}^{q})=\\text{ln}({e}^{p})+\\text{ln}({e}^{q})=p+q=\\text{ln}({e}^{p+q}).[\/latex]<\/div>\r\n<p id=\"fs-id1167793937142\">Since [latex]\\text{ln}x[\/latex] is one-to-one, then<\/p>\r\n\r\n<div id=\"fs-id1167794329135\" class=\"equation unnumbered\">[latex]{e}^{p}{e}^{q}={e}^{p+q}.[\/latex]<\/div>\r\n<p id=\"fs-id1167793503220\">\u25a1<\/p>\r\n<p id=\"fs-id1167793443585\">As with part iv. of the logarithm properties, we can extend property iii. to irrational values of [latex]r,[\/latex] and we do so by the end of the section.<\/p>\r\n<p id=\"fs-id1167793952464\">We also want to verify the differentiation formula for the function [latex]y={e}^{x}.[\/latex] To do this, we need to use implicit differentiation. Let [latex]y={e}^{x}.[\/latex] Then<\/p>\r\n\r\n<div id=\"fs-id1167794095942\" class=\"equation unnumbered\">[latex]\\begin{array}{ccc}\\hfill \\text{ln}y&amp; =\\hfill &amp; x\\hfill \\\\ \\hfill \\frac{d}{dx}\\text{ln}y&amp; =\\hfill &amp; \\frac{d}{dx}x\\hfill \\\\ \\hfill \\frac{1}{y}\\frac{dy}{dx}&amp; =\\hfill &amp; 1\\hfill \\\\ \\hfill \\frac{dy}{dx}&amp; =\\hfill &amp; y.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1167794147337\">Thus, we see<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\frac{d}{dx}{e}^{x}={e}^{x}[\/latex]<\/div>\r\n<p id=\"fs-id1167793370445\">as desired, which leads immediately to the integration formula<\/p>\r\n\r\n<div id=\"fs-id1167793924216\" class=\"equation unnumbered\">[latex]\\int {e}^{x}dx={e}^{x}+C.[\/latex]<\/div>\r\n<p id=\"fs-id1167793272155\">We apply these formulas in the following examples.<\/p>\r\n\r\n<div id=\"fs-id1167793272158\" class=\"textbox examples\">\r\n<h3>Using Properties of Exponential Functions<\/h3>\r\n<div id=\"fs-id1167794098635\" class=\"exercise\">\r\n<div id=\"fs-id1167794098637\" class=\"textbox\">\r\n<p id=\"fs-id1167793393570\">Evaluate the following derivatives:<\/p>\r\n\r\n<ol id=\"fs-id1167793393574\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dt}{e}^{3t}{e}^{{t}^{2}}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}{e}^{3{x}^{2}}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167794293256\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794293256\"]\r\n<p id=\"fs-id1167794293256\">We apply the chain rule as necessary.<\/p>\r\n\r\n<ol id=\"fs-id1167794293259\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dt}{e}^{3t}{e}^{{t}^{2}}=\\frac{d}{dt}{e}^{3t+{t}^{2}}={e}^{3t+{t}^{2}}(3+2t)[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}{e}^{3{x}^{2}}={e}^{3{x}^{2}}6x[\/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-id1167793372329\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1167793442905\" class=\"exercise\">\r\n<div id=\"fs-id1167793442908\" class=\"textbox\">\r\n<p id=\"fs-id1167793374990\">Evaluate the following derivatives:<\/p>\r\n\r\n<ol id=\"fs-id1167793374993\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dx}(\\frac{{e}^{{x}^{2}}}{{e}^{5x}})[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dt}{({e}^{2t})}^{3}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793872351\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793872351\"]\r\n<ol id=\"fs-id1167793872351\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dx}(\\frac{{e}^{{x}^{2}}}{{e}^{5x}})={e}^{{x}^{2}-5x}(2x-5)[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dt}{({e}^{2t})}^{3}=6{e}^{6t}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1167794136708\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1167793568428\">Use the properties of exponential functions and the chain rule as necessary.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793944672\" class=\"textbox examples\">\r\n<h3>Using Properties of Exponential Functions<\/h3>\r\n<div id=\"fs-id1167794031059\" class=\"exercise\">\r\n<div id=\"fs-id1167794031061\" class=\"textbox\">\r\n<p id=\"fs-id1167794020845\">Evaluate the following integral: [latex]\\int 2x{e}^{\\text{\u2212}{x}^{2}}dx.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793501968\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793501968\"]\r\n<p id=\"fs-id1167793501968\">Using [latex]u[\/latex]-substitution, let [latex]u=\\text{\u2212}{x}^{2}.[\/latex] Then [latex]du=-2xdx,[\/latex] and we have<\/p>\r\n\r\n<div id=\"fs-id1167793956578\" class=\"equation unnumbered\">[latex]\\int 2x{e}^{\\text{\u2212}{x}^{2}}dx=\\text{\u2212}\\int {e}^{u}du=\\text{\u2212}{e}^{u}+C=\\text{\u2212}{e}^{\\text{\u2212}{x}^{2}}+C.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793478795\" class=\"textbox exercises checkpoint\">\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1167793553724\">Evaluate the following integral: [latex]\\int \\frac{4}{{e}^{3x}}dx.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793510888\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793510888\"]\r\n<p id=\"fs-id1167793510888\">[latex]\\int \\frac{4}{{e}^{3x}}dx=-\\frac{4}{3}{e}^{-3x}+C[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1167794178061\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1167793943946\">Use the properties of exponential functions and [latex]u\\text{-substitution}[\/latex] as necessary.<\/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-id1167793400840\" class=\"bc-section section\">\r\n<h1>General Logarithmic and Exponential Functions<\/h1>\r\n<p id=\"fs-id1167793400845\">We close this section by looking at exponential functions and logarithms with bases other than [latex]e.[\/latex] Exponential functions are functions of the form [latex]f(x)={a}^{x}.[\/latex] Note that unless [latex]a=e,[\/latex] we still do not have a mathematically rigorous definition of these functions for irrational exponents. Let\u2019s rectify that here by defining the function [latex]f(x)={a}^{x}[\/latex] in terms of the exponential function [latex]{e}^{x}.[\/latex] We then examine logarithms with bases other than [latex]e[\/latex] as inverse functions of exponential functions.<\/p>\r\n\r\n<div id=\"fs-id1167793285142\" class=\"textbox key-takeaways\">\r\n<div class=\"title\">\r\n<h3>Definition<\/h3>\r\n<\/div>\r\nFor any [latex]a&gt;0,[\/latex] and for any real number [latex]x,[\/latex] define [latex]y={a}^{x}[\/latex] as follows:\r\n<div id=\"fs-id1167793420642\" class=\"equation unnumbered\">[latex]y={a}^{x}={e}^{x\\text{ln}a}.[\/latex]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793455076\">Now [latex]{a}^{x}[\/latex] is defined rigorously for all values of [latex]x[\/latex]. This definition also allows us to generalize property iv. of logarithms and property iii. of exponential functions to apply to both rational and irrational values of [latex]r.[\/latex] It is straightforward to show that properties of exponents hold for general exponential functions defined in this way.<\/p>\r\n<p id=\"fs-id1167793384514\">Let\u2019s now apply this definition to calculate a differentiation formula for [latex]{a}^{x}.[\/latex] We have<\/p>\r\n\r\n<div id=\"fs-id1167793559160\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}{a}^{x}=\\frac{d}{dx}{e}^{x\\text{ln}a}={e}^{x\\text{ln}a}\\text{ln}a={a}^{x}\\text{ln}a.[\/latex]<\/div>\r\n<p id=\"fs-id1167794075642\">The corresponding integration formula follows immediately.<\/p>\r\n\r\n<div id=\"fs-id1167794075645\" class=\"textbox key-takeaways theorem\">\r\n<h3>Derivatives and Integrals Involving General Exponential Functions<\/h3>\r\n<p id=\"fs-id1167793293670\">Let [latex]a&gt;0.[\/latex] Then,<\/p>\r\n\r\n<div id=\"fs-id1167793271586\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}{a}^{x}={a}^{x}\\text{ln}a[\/latex]<\/div>\r\n<p id=\"fs-id1167793562027\">and<\/p>\r\n\r\n<div id=\"fs-id1167793562030\" class=\"equation unnumbered\">[latex]\\int {a}^{x}dx=\\frac{1}{\\text{ln}a}{a}^{x}+C.[\/latex]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793267617\">If [latex]a\\ne 1,[\/latex] then the function [latex]{a}^{x}[\/latex] is one-to-one and has a well-defined inverse. Its inverse is denoted by [latex]{\\text{log}}_{a}x.[\/latex] Then,<\/p>\r\n\r\n<div id=\"fs-id1167793776857\" class=\"equation unnumbered\">[latex]y={\\text{log}}_{a}x\\text{if and only if}x={a}^{y}.[\/latex]<\/div>\r\n<p id=\"fs-id1167793929151\">Note that general logarithm functions can be written in terms of the natural logarithm. Let [latex]y={\\text{log}}_{a}x.[\/latex] Then, [latex]x={a}^{y}.[\/latex] Taking the natural logarithm of both sides of this second equation, we get<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\begin{array}{ccc}\\hfill \\text{ln}x&amp; =\\hfill &amp; \\text{ln}({a}^{y})\\hfill \\\\ \\hfill \\text{ln}x&amp; =\\hfill &amp; y\\text{ln}a\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; \\frac{\\text{ln}x}{\\text{ln}a}\\hfill \\\\ \\hfill {\\text{log}}_{}x&amp; =\\hfill &amp; \\frac{\\text{ln}x}{\\text{ln}a}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1167793450615\">Thus, we see that all logarithmic functions are constant multiples of one another. Next, we use this formula to find a differentiation formula for a logarithm with base [latex]a.[\/latex] Again, let [latex]y={\\text{log}}_{a}x.[\/latex] Then,<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill \\frac{dy}{dx}&amp; =\\frac{d}{dx}({\\text{log}}_{a}x)\\hfill \\\\ &amp; =\\frac{d}{dx}(\\frac{\\text{ln}x}{\\text{ln}a})\\hfill \\\\ &amp; =(\\frac{1}{\\text{ln}a})\\frac{d}{dx}(\\text{ln}x)\\hfill \\\\ &amp; =\\frac{1}{\\text{ln}a}\u00b7\\frac{1}{x}\\hfill \\\\ &amp; =\\frac{1}{x\\text{ln}a}.\\hfill \\end{array}[\/latex]<\/div>\r\n<div id=\"fs-id1167794324570\" class=\"textbox key-takeaways theorem\">\r\n<h3>Derivatives of General Logarithm Functions<\/h3>\r\n<p id=\"fs-id1167793432748\">Let [latex]a&gt;0.[\/latex] Then,<\/p>\r\n\r\n<div id=\"fs-id1167794139845\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}{\\text{log}}_{a}x=\\frac{1}{x\\text{ln}a}.[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<div id=\"fs-id1167793948186\" class=\"exercise\">\r\n<div id=\"fs-id1167793948188\" class=\"textbox\">\r\n<h3>Calculating Derivatives of General Exponential and Logarithm Functions<\/h3>\r\n<p id=\"fs-id1167793640049\">Evaluate the following derivatives:<\/p>\r\n\r\n<ol id=\"fs-id1167793640052\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dt}({4}^{t}\u00b7{2}^{{t}^{2}})[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}{\\text{log}}_{8}(7{x}^{2}+4)[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793298214\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793298214\"]\r\n<p id=\"fs-id1167793298214\">We need to apply the chain rule as necessary.<\/p>\r\n\r\n<ol id=\"fs-id1167793829825\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dt}({4}^{t}\u00b7{2}^{{t}^{2}})=\\frac{d}{dt}({2}^{2t}\u00b7{2}^{{t}^{2}})=\\frac{d}{dt}({2}^{2t+{t}^{2}})={2}^{2t+{t}^{2}}\\text{ln}(2)(2+2t)[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}{\\text{log}}_{8}(7{x}^{2}+4)=\\frac{1}{(7{x}^{2}+4)(\\text{ln}8)}(14x)[\/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-id1167793455288\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1167793455291\" class=\"exercise\">\r\n<div id=\"fs-id1167793455293\" class=\"textbox\">\r\n<p id=\"fs-id1167793543534\">Evaluate the following derivatives:<\/p>\r\n\r\n<ol style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dt}{4}^{{t}^{4}}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}{\\text{log}}_{3}(\\sqrt{{x}^{2}+1})[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1167793978416\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793978416\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793978416\"]\r\n<ol style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dt}{4}^{{t}^{4}}={4}^{{t}^{4}}(\\text{ln}4)(4{t}^{3})[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}{\\text{log}}_{3}(\\sqrt{{x}^{2}+1})=\\frac{x}{(\\text{ln}3)({x}^{2}+1)}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1167793521316\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\nUse the formulas and apply the chain rule as necessary.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793499138\" class=\"textbox examples\">\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n<h3>Integrating General Exponential Functions<\/h3>\r\n<p id=\"fs-id1167793499148\">Evaluate the following integral: [latex]\\int \\frac{3}{{2}^{3x}}dx.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793956537\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793956537\"]\r\n<p id=\"fs-id1167793956537\">Use [latex]u\\text{-substitution}[\/latex] and let [latex]u=-3x.[\/latex] Then [latex]du=-3dx[\/latex] and we have<\/p>\r\n\r\n<div id=\"fs-id1167793929418\" class=\"equation unnumbered\">[latex]\\int \\frac{3}{{2}^{3x}}dx=\\int 3\u00b7{2}^{-3x}dx=\\text{\u2212}\\int {2}^{u}du=-\\frac{1}{\\text{ln}2}{2}^{u}+C=-\\frac{1}{\\text{ln}2}{2}^{-3x}+C.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793789604\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1167793789608\" class=\"exercise\">\r\n<div id=\"fs-id1167794188287\" class=\"textbox\">\r\n<p id=\"fs-id1167794188289\">Evaluate the following integral: [latex]\\int {x}^{2}{2}^{{x}^{3}}dx.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793979123\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793979123\"]\r\n<p id=\"fs-id1167793979123\">[latex]\\int {x}^{2}{2}^{{x}^{3}}dx=\\frac{1}{3\\text{ln}2}{2}^{{x}^{3}}+C[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1167793638224\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1167793423554\">Use the properties of exponential functions and [latex]u\\text{-substitution}[\/latex] as necessary.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793423569\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1167793931233\">\r\n \t<li>The earlier treatment of logarithms and exponential functions did not define the functions precisely and formally. This section develops the concepts in a mathematically rigorous way.<\/li>\r\n \t<li>The cornerstone of the development is the definition of the natural logarithm in terms of an integral.<\/li>\r\n \t<li>The function [latex]{e}^{x}[\/latex] is then defined as the inverse of the natural logarithm.<\/li>\r\n \t<li>General exponential functions are defined in terms of [latex]{e}^{x},[\/latex] and the corresponding inverse functions are general logarithms.<\/li>\r\n \t<li>Familiar properties of logarithms and exponents still hold in this more rigorous context.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1167794327228\" class=\"key-equations\">\r\n<h1>Key Equations<\/h1>\r\n<ul id=\"fs-id1167793278406\">\r\n \t<li><strong>Natural logarithm function<\/strong><\/li>\r\n \t<li>[latex]\\text{ln}x={\\int }_{1}^{x}\\frac{1}{t}dt[\/latex] Z<\/li>\r\n \t<li><strong>Exponential function<\/strong>[latex]y={e}^{x}[\/latex]<\/li>\r\n \t<li>[latex]\\text{ln}y=\\text{ln}({e}^{x})=x[\/latex] Z<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1167793879358\" class=\"textbox exercises\">\r\n<p id=\"fs-id1167793879362\">For the following exercises, find the derivative [latex]\\frac{dy}{dx}.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1167793363071\" class=\"exercise\">\r\n<div id=\"fs-id1167793363073\" class=\"textbox\">\r\n<p id=\"fs-id1167793363076\">[latex]y=\\text{ln}(2x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167794097576\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794097576\"]\r\n<p id=\"fs-id1167794097576\">[latex]\\frac{1}{x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794097588\" class=\"exercise\">\r\n<div id=\"fs-id1167794226012\" class=\"textbox\">\r\n<p id=\"fs-id1167794226014\">[latex]y=\\text{ln}(2x+1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793831549\" class=\"exercise\">\r\n<div id=\"fs-id1167793629438\" class=\"textbox\">\r\n<p id=\"fs-id1167793629440\">[latex]y=\\frac{1}{\\text{ln}x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793546866\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793546866\"]\r\n<p id=\"fs-id1167793546866\">[latex]-\\frac{1}{x{(\\text{ln}x)}^{2}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793541105\">For the following exercises, find the indefinite integral.<\/p>\r\n\r\n<div id=\"fs-id1167793541108\" class=\"exercise\">\r\n<div id=\"fs-id1167793541110\" class=\"textbox\">\r\n<p id=\"fs-id1167793541112\">[latex]\\int \\frac{dt}{3t}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793367825\" class=\"exercise\">\r\n<div id=\"fs-id1167793367827\" class=\"textbox\">\r\n<p id=\"fs-id1167793367829\">[latex]\\int \\frac{dx}{1+x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n\r\n[latex]\\text{ln}(x+1)+C[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793706099\">For the following exercises, find the derivative [latex]dy\\text{\/}dx.[\/latex] (You can use a calculator to plot the function and the derivative to confirm that it is correct.)<\/p>\r\n\r\n<div id=\"fs-id1167793706119\" class=\"exercise\">\r\n<div id=\"fs-id1167793706121\" class=\"textbox\">\r\n<p id=\"fs-id1167793706123\"><strong>[T]<\/strong>[latex]y=\\frac{\\text{ln}(x)}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793510876\" class=\"exercise\">\r\n<div id=\"fs-id1167793510878\" class=\"textbox\">\r\n<p id=\"fs-id1167793287409\"><strong>[T]<\/strong>[latex]y=x\\text{ln}(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<p id=\"fs-id1167793628620\">[latex]\\text{ln}(x)+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793628640\" class=\"exercise\">\r\n<div id=\"fs-id1167793628642\" class=\"textbox\">\r\n<p id=\"fs-id1167793628644\"><strong>[T]<\/strong>[latex]y={\\text{log}}_{10}x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793244348\" class=\"exercise\">\r\n<div id=\"fs-id1167793244350\" class=\"textbox\">\r\n<p id=\"fs-id1167793244352\"><strong>[T]<\/strong>[latex]y=\\text{ln}( \\sin x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793776894\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793776894\"]\r\n<p id=\"fs-id1167793776894\">[latex] \\cot (x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793504037\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1167793504041\"><strong>[T]<\/strong>[latex]y=\\text{ln}(\\text{ln}x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793423443\" class=\"exercise\">\r\n<div id=\"fs-id1167793423445\" class=\"textbox\">\r\n<p id=\"fs-id1167793423447\"><strong>[T]<\/strong>[latex]y=7\\text{ln}(4x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793293693\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793293693\"]\r\n<p id=\"fs-id1167793293693\">[latex]\\frac{7}{x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793293705\" class=\"exercise\">\r\n<div id=\"fs-id1167793293707\" class=\"textbox\">\r\n<p id=\"fs-id1167793293709\"><strong>[T]<\/strong>[latex]y=\\text{ln}({(4x)}^{7})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794073092\" class=\"exercise\">\r\n<div id=\"fs-id1167794073094\" class=\"textbox\">\r\n<p id=\"fs-id1167794073096\"><strong>[T]<\/strong>[latex]y=\\text{ln}( \\tan x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793557826\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793557826\"]\r\n<p id=\"fs-id1167793557826\">[latex] \\csc (x) \\sec x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793563636\" class=\"exercise\">\r\n<div id=\"fs-id1167793563639\" class=\"textbox\">\r\n<p id=\"fs-id1167793563641\"><strong>[T]<\/strong>[latex]y=\\text{ln}( \\tan (3x))[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1167793281524\" class=\"textbox\">\r\n<p id=\"fs-id1167793281526\"><strong>[T]<\/strong>[latex]y=\\text{ln}({ \\cos }^{2}x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n\r\n[latex]-2 \\tan x[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793553757\">For the following exercises, find the definite or indefinite integral.<\/p>\r\n\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n\r\n[latex]{\\int }_{0}^{1}\\frac{dx}{3+x}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793372398\" class=\"exercise\">\r\n<div id=\"fs-id1167793372400\" class=\"textbox\">\r\n<p id=\"fs-id1167793372402\">[latex]{\\int }_{0}^{1}\\frac{dt}{3+2t}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793553651\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793553651\"]\r\n<p id=\"fs-id1167793553651\">[latex]\\frac{1}{2}\\text{ln}(\\frac{5}{3})[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793582462\" class=\"exercise\">\r\n<div id=\"fs-id1167793582464\" class=\"textbox\">\r\n<p id=\"fs-id1167793582466\">[latex]{\\int }_{0}^{2}\\frac{xdx}{{x}^{2}+1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794122084\" class=\"exercise\">\r\n<div id=\"fs-id1167794122086\" class=\"textbox\">\r\n<p id=\"fs-id1167794122088\">[latex]{\\int }_{0}^{2}\\frac{{x}^{3}dx}{{x}^{2}+1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793952145\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793952145\"]\r\n<p id=\"fs-id1167793952145\">[latex]2-\\frac{1}{2}\\text{ln}(5)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793499058\" class=\"exercise\">\r\n<div id=\"fs-id1167793499060\" class=\"textbox\">\r\n<p id=\"fs-id1167793499062\">[latex]{\\int }_{2}^{e}\\frac{dx}{x\\text{ln}x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794054231\" class=\"exercise\">\r\n<div id=\"fs-id1167794054233\" class=\"textbox\">\r\n<p id=\"fs-id1167794054235\">[latex]{\\int }_{2}^{e}\\frac{dx}{{(x\\text{ln}(x))}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<p id=\"fs-id1167793473569\">[latex]\\frac{1}{\\text{ln}(2)}-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793481974\" class=\"exercise\">\r\n<div id=\"fs-id1167793481976\" class=\"textbox\">\r\n<p id=\"fs-id1167793481978\">[latex]\\int \\frac{ \\cos xdx}{ \\sin x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793469786\" class=\"exercise\">\r\n<div id=\"fs-id1167793469788\" class=\"textbox\">\r\n<p id=\"fs-id1167793469791\">[latex]{\\int }_{0}^{\\pi \\text{\/}4} \\tan xdx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1167794005215\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1167794005215\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794005215\"][latex]\\frac{1}{2}\\text{ln}(2)[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794005239\" class=\"exercise\">\r\n<div id=\"fs-id1167794005241\" class=\"textbox\">\r\n<p id=\"fs-id1167794005243\">[latex]\\int \\cot (3x)dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794291581\" class=\"exercise\">\r\n<div id=\"fs-id1167794291583\" class=\"textbox\">\r\n<p id=\"fs-id1167794291585\">[latex]\\int \\frac{{(\\text{ln}x)}^{2}dx}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793421199\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793421199\"]\r\n<p id=\"fs-id1167793421199\">[latex]\\frac{1}{3}{(\\text{ln}x)}^{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nFor the following exercises, compute [latex]dy\\text{\/}dx[\/latex] by differentiating [latex]\\text{ln}y.[\/latex]\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n\r\n[latex]y=\\sqrt{{x}^{2}+1}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793394990\" class=\"exercise\">\r\n<div id=\"fs-id1167793394992\" class=\"textbox\">\r\n<p id=\"fs-id1167793394994\">[latex]y=\\sqrt{{x}^{2}+1}\\sqrt{{x}^{2}-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793595181\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793595181\"]\r\n<p id=\"fs-id1167793595181\">[latex]\\frac{2{x}^{3}}{\\sqrt{{x}^{2}+1}\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793595222\" class=\"exercise\">\r\n<div id=\"fs-id1167793595224\" class=\"textbox\">\r\n<p id=\"fs-id1167793595226\">[latex]y={e}^{ \\sin x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793634250\" class=\"exercise\">\r\n<div id=\"fs-id1167793634252\" class=\"textbox\">\r\n<p id=\"fs-id1167793634254\">[latex]y={x}^{-1\\text{\/}x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793465231\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793465231\"]\r\n<p id=\"fs-id1167793465231\">[latex]{x}^{-2-(1\\text{\/}x)}(\\text{ln}x-1)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793465279\" class=\"exercise\">\r\n<div id=\"fs-id1167793465281\" class=\"textbox\">\r\n<p id=\"fs-id1167793465283\">[latex]y={e}^{(ex)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n\r\n[latex]y={x}^{e}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793445732\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793445732\"]\r\n<p id=\"fs-id1167793445732\">[latex]e{x}^{e-1}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793445751\" class=\"exercise\">\r\n<div id=\"fs-id1167793445753\" class=\"textbox\">\r\n<p id=\"fs-id1167793445755\">[latex]y={x}^{(ex)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793543331\" class=\"exercise\">\r\n<div id=\"fs-id1167793543333\" class=\"textbox\">\r\n<p id=\"fs-id1167793543335\">[latex]y=\\sqrt{x}\\sqrt[3]{x}\\sqrt[6]{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793566015\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793566015\"]\r\n<p id=\"fs-id1167793566015\">1<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793566023\" class=\"exercise\">\r\n<div id=\"fs-id1167793566025\" class=\"textbox\">\r\n<p id=\"fs-id1167793566027\">[latex]y={x}^{-1\\text{\/}\\text{ln}x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793566062\" class=\"exercise\">\r\n<div id=\"fs-id1167793566064\" class=\"textbox\">\r\n<p id=\"fs-id1167793566066\">[latex]y={e}^{\\text{\u2212}\\text{ln}x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793315539\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793315539\"]\r\n<p id=\"fs-id1167793315539\">[latex]-\\frac{1}{{x}^{2}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793315557\">For the following exercises, evaluate by any method.<\/p>\r\n\r\n<div id=\"fs-id1167793315561\" class=\"exercise\">\r\n<div id=\"fs-id1167793315563\" class=\"textbox\">\r\n<p id=\"fs-id1167793315565\">[latex]{\\int }_{5}^{10}\\frac{dt}{t}-{\\int }_{5x}^{10x}\\frac{dt}{t}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793937978\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n\r\n[latex]{\\int }_{1}^{{e}^{\\pi }}\\frac{dx}{x}+{\\int }_{-2}^{-1}\\frac{dx}{x}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793455346\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793455346\"]\r\n<p id=\"fs-id1167793455346\">[latex]\\pi -\\text{ln}(2)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1167794127470\" class=\"textbox\">\r\n<p id=\"fs-id1167794127472\">[latex]\\frac{d}{dx}{\\int }_{x}^{1}\\frac{dt}{t}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794127523\" class=\"exercise\">\r\n<div id=\"fs-id1167794127525\" class=\"textbox\">\r\n<p id=\"fs-id1167794127527\">[latex]\\frac{d}{dx}{\\int }_{x}^{{x}^{2}}\\frac{dt}{t}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167794146824\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794146824\"]\r\n<p id=\"fs-id1167794146824\">[latex]\\frac{1}{x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794146835\" class=\"exercise\">\r\n<div id=\"fs-id1167794146838\" class=\"textbox\">\r\n<p id=\"fs-id1167794146840\">[latex]\\frac{d}{dx}\\text{ln}( \\sec x+ \\tan x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793423282\">For the following exercises, use the function [latex]\\text{ln}x.[\/latex] If you are unable to find intersection points analytically, use a calculator.<\/p>\r\n\r\n<div id=\"fs-id1167793423297\" class=\"exercise\">\r\n<div id=\"fs-id1167793423299\" class=\"textbox\">\r\n<p id=\"fs-id1167793423301\">Find the area of the region enclosed by [latex]x=1[\/latex] and [latex]y=5[\/latex] above [latex]y=\\text{ln}x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793541846\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793541846\"]\r\n<p id=\"fs-id1167793541846\">[latex]{e}^{5}-6{\\text{units}}^{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793541870\" class=\"exercise\">\r\n<div id=\"fs-id1167793541872\" class=\"textbox\">\r\n<p id=\"fs-id1167793541874\"><strong>[T]<\/strong> Find the arc length of [latex]\\text{ln}x[\/latex] from [latex]x=1[\/latex] to [latex]x=2.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793454016\" class=\"exercise\">\r\n<div id=\"fs-id1167793454018\" class=\"textbox\">\r\n<p id=\"fs-id1167793454020\">Find the area between [latex]\\text{ln}x[\/latex] and the [latex]x[\/latex]-axis from [latex]x=1\\text{ to }x=2.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793713050\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793713050\"]\r\n<p id=\"fs-id1167793713050\">[latex]\\text{ln}(4)-1{\\text{units}}^{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793713080\" class=\"exercise\">\r\n<div id=\"fs-id1167793713082\" class=\"textbox\">\r\n<p id=\"fs-id1167793713084\">Find the volume of the shape created when rotating this curve from [latex]x=1\\text{ to }x=2[\/latex] around the [latex]x[\/latex]-axis, as pictured here.<\/p>\r\n<span id=\"fs-id1167793960058\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213332\/CNX_Calc_Figure_06_07_201.jpg\" alt=\"This figure is a surface. It has been generated by revolving the curve ln x about the x-axis. The surface is inside of a cube showing it is 3-dimensinal.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793510646\" class=\"exercise\">\r\n<div id=\"fs-id1167793510649\" class=\"textbox\">\r\n<p id=\"fs-id1167793510651\"><strong>[T]<\/strong> Find the surface area of the shape created when rotating the curve in the previous exercise from [latex]x=1[\/latex] to [latex]x=2[\/latex] around the [latex]x[\/latex]-axis.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793510687\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793510687\"]\r\n<p id=\"fs-id1167793510687\">2.8656<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793510695\">If you are unable to find intersection points analytically in the following exercises, use a calculator.<\/p>\r\n\r\n<div id=\"fs-id1167793510699\" class=\"exercise\">\r\n<div id=\"fs-id1167793510702\" class=\"textbox\">\r\n<p id=\"fs-id1167793510704\">Find the area of the hyperbolic quarter-circle enclosed by [latex]x=2\\text{ and }y=2[\/latex] above [latex]y=1\\text{\/}x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794209645\" class=\"exercise\">\r\n<div id=\"fs-id1167794209648\" class=\"textbox\">\r\n<p id=\"fs-id1167794209650\"><strong>[T]<\/strong> Find the arc length of [latex]y=1\\text{\/}x[\/latex] from [latex]x=1\\text{ to }x=4.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793570742\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793570742\"]\r\n<p id=\"fs-id1167793570742\">3.1502<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793570750\" class=\"exercise\">\r\n<div id=\"fs-id1167793570752\" class=\"textbox\">\r\n<p id=\"fs-id1167793570755\">Find the area under [latex]y=1\\text{\/}x[\/latex] and above the [latex]x[\/latex]-axis from [latex]x=1\\text{ to }x=4.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793593495\">For the following exercises, verify the derivatives and antiderivatives.<\/p>\r\n\r\n<div id=\"fs-id1167793593498\" class=\"exercise\">\r\n<div id=\"fs-id1167793593500\" class=\"textbox\">\r\n<p id=\"fs-id1167793593502\">[latex]\\frac{d}{dx}\\text{ln}(x+\\sqrt{{x}^{2}+1})=\\frac{1}{\\sqrt{1+{x}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793377930\" class=\"exercise\">\r\n<div id=\"fs-id1167793377932\" class=\"textbox\">\r\n<p id=\"fs-id1167793377934\">[latex]\\frac{d}{dx}\\text{ln}(\\frac{x-a}{x+a})=\\frac{2a}{({x}^{2}-{a}^{2})}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793380054\" class=\"exercise\">\r\n<div id=\"fs-id1167793380056\" class=\"textbox\">\r\n<p id=\"fs-id1167793380058\">[latex]\\frac{d}{dx}\\text{ln}(\\frac{1+\\sqrt{1-{x}^{2}}}{x})=-\\frac{1}{x\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793911914\" class=\"exercise\">\r\n<div id=\"fs-id1167793911916\" class=\"textbox\">\r\n<p id=\"fs-id1167793911918\">[latex]\\frac{d}{dx}\\text{ln}(x+\\sqrt{{x}^{2}-{a}^{2}})=\\frac{1}{\\sqrt{{x}^{2}-{a}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793373565\" class=\"exercise\">\r\n<div id=\"fs-id1167793373567\" class=\"textbox\">\r\n<p id=\"fs-id1167793373569\">[latex]\\int \\frac{dx}{x\\text{ln}(x)\\text{ln}(\\text{ln}x)}=\\text{ln}(\\text{ln}(\\text{ln}x))+C[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Write the definition of the natural logarithm as an integral.<\/li>\n<li>Recognize the derivative of the natural logarithm.<\/li>\n<li>Integrate functions involving the natural logarithmic function.<\/li>\n<li>Define the number [latex]e[\/latex] through an integral.<\/li>\n<li>Recognize the derivative and integral of the exponential function.<\/li>\n<li>Prove properties of logarithms and exponential functions using integrals.<\/li>\n<li>Express general logarithmic and exponential functions in terms of natural logarithms and exponentials.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1167793957648\">We already examined exponential functions and logarithms in earlier chapters. However, we glossed over some key details in the previous discussions. For example, we did not study how to treat exponential functions with exponents that are irrational. The definition of the number [latex]e[\/latex] is another area where the previous development was somewhat incomplete. We now have the tools to deal with these concepts in a more mathematically rigorous way, and we do so in this section.<\/p>\n<p id=\"fs-id1167793417231\">For purposes of this section, assume we have not yet defined the natural logarithm, the number [latex]e[\/latex], or any of the integration and differentiation formulas associated with these functions. By the end of the section, we will have studied these concepts in a mathematically rigorous way (and we will see they are consistent with the concepts we learned earlier).<\/p>\n<p>We begin the section by defining the natural logarithm in terms of an integral. This definition forms the foundation for the section. From this definition, we derive differentiation formulas, define the number [latex]e,[\/latex] and expand these concepts to logarithms and exponential functions of any base.<\/p>\n<div id=\"fs-id1167794054166\" class=\"bc-section section\">\n<h1>The Natural Logarithm as an Integral<\/h1>\n<p id=\"fs-id1167793958127\">Recall the power rule for integrals:<\/p>\n<div id=\"fs-id1167793299462\" class=\"equation unnumbered\">[latex]\\int {x}^{n}dx=\\frac{{x}^{n+1}}{n+1}+C,n\\ne \\text{\u2212}1.[\/latex]<\/div>\n<p>Clearly, this does not work when [latex]n=-1,[\/latex] as it would force us to divide by zero. So, what do we do with [latex]\\int \\frac{1}{x}dx?[\/latex] Recall from the Fundamental Theorem of Calculus that [latex]{\\int }_{1}^{x}\\frac{1}{t}dt[\/latex] is an antiderivative of [latex]1\\text{\/}x.[\/latex] Therefore, we can make the following definition.<\/p>\n<div id=\"fs-id1167793370244\" class=\"textbox key-takeaways\">\n<div class=\"title\">\n<h3>Definition<\/h3>\n<\/div>\n<p id=\"fs-id1167793953542\">For [latex]x>0,[\/latex] define the natural logarithm function by<\/p>\n<div id=\"fs-id1167793550098\" class=\"equation\">[latex]\\text{ln}x={\\int }_{1}^{x}\\frac{1}{t}dt.[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1167793617974\">For [latex]x>1,[\/latex] this is just the area under the curve [latex]y=1\\text{\/}t[\/latex] from 1 to [latex]x.[\/latex] For [latex]x<1,[\/latex] we have [latex]{\\int }_{1}^{x}\\frac{1}{t}dt=\\text{\u2212}{\\int }_{x}^{1}\\frac{1}{t}dt,[\/latex] so in this case it is the negative of the area under the curve from [latex]x\\text{ to }1[\/latex] (see the following figure).<\/p>\n<div id=\"CNX_Calc_Figure_06_07_001\" class=\"wp-caption aligncenter\">\n<div style=\"width: 599px\" 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\/11213320\/CNX_Calc_Figure_06_07_001.jpg\" alt=\"This figure has two graphs. The first is the curve y=1\/t. It is decreasing and in the first quadrant. Under the curve is a shaded area. The area is bounded to the left at x=1. The area is labeled \u201carea=lnx\u201d. The second graph is the same curve y=1\/t. It has shaded area under the curve bounded to the right by x=1. It is labeled \u201carea=-lnx\u201d.\" width=\"589\" height=\"311\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. (a) When [latex]x&gt;1,[\/latex] the natural logarithm is the area under the curve [latex]y=1\\text{\/}t[\/latex] from [latex]1\\text{ to }x.[\/latex] (b) When [latex]x&lt;1,[\/latex] the natural logarithm is the negative of the area under the curve from [latex]x[\/latex] to 1.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793932688\">Notice that [latex]\\text{ln}1=0.[\/latex] Furthermore, the function [latex]y=1\\text{\/}t>0[\/latex] for [latex]x>0.[\/latex] Therefore, by the properties of integrals, it is clear that [latex]\\text{ln}x[\/latex] is increasing for [latex]x>0.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167793930384\" class=\"bc-section section\">\n<h1>Properties of the Natural Logarithm<\/h1>\n<p id=\"fs-id1167794139011\">Because of the way we defined the natural logarithm, the following differentiation formula falls out immediately as a result of to the Fundamental Theorem of Calculus.<\/p>\n<div id=\"fs-id1167794186831\" class=\"textbox key-takeaways theorem\">\n<h3>Derivative of the Natural Logarithm<\/h3>\n<p id=\"fs-id1167794050615\">For [latex]x>0,[\/latex] the derivative of the natural logarithm is given by<\/p>\n<div id=\"fs-id1167793432137\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}\\text{ln}x=\\frac{1}{x}.[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167793879890\" class=\"textbox key-takeaways theorem\">\n<h3>Corollary to the Derivative of the Natural Logarithm<\/h3>\n<p id=\"fs-id1167794128288\">The function [latex]\\text{ln}x[\/latex] is differentiable; therefore, it is continuous.<\/p>\n<\/div>\n<p id=\"fs-id1167794141119\">A graph of [latex]\\text{ln}x[\/latex] is shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_06_07_002\">(Figure)<\/a>. Notice that it is continuous throughout its domain of [latex](0,\\infty ).[\/latex]<\/p>\n<div id=\"CNX_Calc_Figure_06_07_002\" class=\"wp-caption aligncenter\">\n<div style=\"width: 276px\" 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\/11213322\/CNX_Calc_Figure_06_07_002.jpg\" alt=\"This figure is a graph. It is an increasing curve labeled f(x)=lnx. The curve is increasing with the y-axis as an asymptote. The curve intersects the x-axis at x=1.\" width=\"266\" height=\"347\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. The graph of [latex]f(x)=\\text{ln}x[\/latex] shows that it is a continuous function.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793967054\" class=\"textbox examples\">\n<h3>Calculating Derivatives of Natural Logarithms<\/h3>\n<div id=\"fs-id1167793424342\" class=\"exercise\">\n<div id=\"fs-id1167793425720\" class=\"textbox\">\n<p id=\"fs-id1167794168088\">Calculate the following derivatives:<\/p>\n<ol id=\"fs-id1167793966989\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dx}\\text{ln}(5{x}^{3}-2)[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{(\\text{ln}(3x))}^{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-id1167794171377\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167794171377\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794171377\">We need to apply the chain rule in both cases.<\/p>\n<ol id=\"fs-id1167793926297\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dx}\\text{ln}(5{x}^{3}-2)=\\frac{15{x}^{2}}{5{x}^{3}-2}[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{(\\text{ln}(3x))}^{2}=\\frac{2(\\text{ln}(3x))\u00b73}{3x}=\\frac{2(\\text{ln}(3x))}{x}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793398778\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1167793369601\" class=\"exercise\">\n<div id=\"fs-id1167794052465\" class=\"textbox\">\n<p id=\"fs-id1167794334473\">Calculate the following derivatives:<\/p>\n<ol id=\"fs-id1167793964608\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dx}\\text{ln}(2{x}^{2}+x)[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{(\\text{ln}({x}^{3}))}^{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-id1167793259644\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793259644\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1167793259644\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dx}\\text{ln}(2{x}^{2}+x)=\\frac{4x+1}{2{x}^{2}+x}[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{(\\text{ln}({x}^{3}))}^{2}=\\frac{6\\text{ln}({x}^{3})}{x}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1167793277086\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1167794187630\">Apply the differentiation formula just provided and use the chain rule as necessary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793609503\">Note that if we use the absolute value function and create a new function [latex]\\text{ln}|x|,[\/latex] we can extend the domain of the natural logarithm to include [latex]x<0.[\/latex] Then [latex](d\\text{\/}(dx))\\text{ln}|x|=1\\text{\/}x.[\/latex] This gives rise to the familiar integration formula.<\/p>\n<div id=\"fs-id1167793432204\" class=\"textbox key-takeaways theorem\">\n<h3>Integral of (1\/[latex]u[\/latex]) <em>du<\/em><\/h3>\n<p id=\"fs-id1167793832055\">The natural logarithm is the antiderivative of the function [latex]f(u)=1\\text{\/}u\\text{:}[\/latex]<\/p>\n<div id=\"fs-id1167793973017\" class=\"equation unnumbered\">[latex]\\int \\frac{1}{u}du=\\text{ln}|u|+C.[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167793423035\" class=\"textbox examples\">\n<h3>Calculating Integrals Involving Natural Logarithms<\/h3>\n<div id=\"fs-id1167793948151\" class=\"exercise\">\n<div id=\"fs-id1167794027994\" class=\"textbox\">\n<p id=\"fs-id1167793989365\">Calculate the integral [latex]\\int \\frac{x}{{x}^{2}+4}dx.[\/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-id1167793887223\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793887223\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793887223\">Using [latex]u[\/latex]-substitution, let [latex]u={x}^{2}+4.[\/latex] Then [latex]du=2xdx[\/latex] and we have<\/p>\n<div id=\"fs-id1167793971678\" class=\"equation unnumbered\">[latex]\\int \\frac{x}{{x}^{2}+4}dx=\\frac{1}{2}\\int \\frac{1}{u}du\\frac{1}{2}\\text{ln}|u|+C=\\frac{1}{2}\\text{ln}|{x}^{2}+4|+C=\\frac{1}{2}\\text{ln}({x}^{2}+4)+C.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794041523\" class=\"textbox exercises checkpoint\">\n<div class=\"exercise\">\n<div id=\"fs-id1167793377877\" class=\"textbox\">\n<p id=\"fs-id1167793414202\">Calculate the integral [latex]\\int \\frac{{x}^{2}}{{x}^{3}+6}dx.[\/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-id1167793525036\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793525036\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793525036\">[latex]\\int \\frac{{x}^{2}}{{x}^{3}+6}dx=\\frac{1}{3}\\text{ln}|{x}^{3}+6|+C[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167793549644\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1167793862633\">Apply the integration formula provided earlier and use [latex]u[\/latex]-substitution as necessary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167794062403\">Although we have called our function a \u201clogarithm,\u201d we have not actually proved that any of the properties of logarithms hold for this function. We do so here.<\/p>\n<div id=\"fs-id1167793366787\" class=\"textbox key-takeaways theorem\">\n<h3>Properties of the Natural Logarithm<\/h3>\n<p id=\"fs-id1167793936147\">If [latex]a,b>0[\/latex] and [latex]r[\/latex] is a rational number, then<\/p>\n<ol id=\"fs-id1167793271023\">\n<li>[latex]\\text{ln}1=0[\/latex]<\/li>\n<li>[latex]\\text{ln}(ab)=\\text{ln}a+\\text{ln}b[\/latex]<\/li>\n<li>[latex]\\text{ln}(\\frac{a}{b})=\\text{ln}a-\\text{ln}b[\/latex]<\/li>\n<li>[latex]\\text{ln}({a}^{r})=r\\text{ln}a[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1167793509941\" class=\"bc-section section\">\n<h2>Proof<\/h2>\n<ol id=\"fs-id1167793977078\">\n<li>By definition, [latex]\\text{ln}1={\\int }_{1}^{1}\\frac{1}{t}dt=0.[\/latex]<\/li>\n<li>We have\n<div id=\"fs-id1167793964775\" class=\"equation unnumbered\">[latex]\\text{ln}(ab)={\\int }_{1}^{ab}\\frac{1}{t}dt={\\int }_{1}^{a}\\frac{1}{t}dt+{\\int }_{a}^{ab}\\frac{1}{t}dt.[\/latex]<\/div>\n<p>Use [latex]u\\text{-substitution}[\/latex] on the last integral in this expression. Let [latex]u=t\\text{\/}a.[\/latex] Then [latex]du=(1\\text{\/}a)dt.[\/latex] Furthermore, when [latex]t=a,u=1,[\/latex] and when [latex]t=ab,u=b.[\/latex] So we get<\/p>\n<div id=\"fs-id1167793490878\" class=\"equation unnumbered\">[latex]\\text{ln}(ab)={\\int }_{1}^{a}\\frac{1}{t}dt+{\\int }_{a}^{ab}\\frac{1}{t}dt={\\int }_{1}^{a}\\frac{1}{t}dt+{\\int }_{1}^{ab}\\frac{a}{t}\u00b7\\frac{1}{a}dt={\\int }_{1}^{a}\\frac{1}{t}dt+{\\int }_{1}^{b}\\frac{1}{u}du=\\text{ln}a+\\text{ln}b.[\/latex]<\/div>\n<\/li>\n<li>Note that\n<div id=\"fs-id1167793951598\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}\\text{ln}({x}^{r})=\\frac{r{x}^{r-1}}{{x}^{r}}=\\frac{r}{x}.[\/latex]<\/div>\n<p>Furthermore,<\/p>\n<div class=\"equation unnumbered\">[latex]\\frac{d}{dx}(r\\text{ln}x)=\\frac{r}{x}.[\/latex]<\/div>\n<p>Since the derivatives of these two functions are the same, by the Fundamental Theorem of Calculus, they must differ by a constant. So we have<\/p>\n<div id=\"fs-id1167793563854\" class=\"equation unnumbered\">[latex]\\text{ln}({x}^{r})=r\\text{ln}x+C[\/latex]<\/div>\n<p>for some constant [latex]C.[\/latex] Taking [latex]x=1,[\/latex] we get<\/p>\n<div id=\"fs-id1167793510762\" class=\"equation unnumbered\">[latex]\\begin{array}{ccc}\\hfill \\text{ln}({1}^{r})& =\\hfill & r\\text{ln}(1)+C\\hfill \\\\ \\hfill 0& =\\hfill & r(0)+C\\hfill \\\\ \\hfill C& =\\hfill & 0.\\hfill \\end{array}[\/latex]<\/div>\n<p>Thus [latex]\\text{ln}({x}^{r})=r\\text{ln}x[\/latex] and the proof is complete. Note that we can extend this property to irrational values of [latex]r[\/latex] later in this section.<br \/>\nPart iii. follows from parts ii. and iv. and the proof is left to you.<\/li>\n<\/ol>\n<p id=\"fs-id1167794210729\">\u25a1<\/p>\n<div class=\"textbox examples\">\n<div id=\"fs-id1167794077125\" class=\"exercise\">\n<div id=\"fs-id1167793961386\" class=\"textbox\">\n<h3>Using Properties of Logarithms<\/h3>\n<p id=\"fs-id1167794003628\">Use properties of logarithms to simplify the following expression into a single logarithm:<\/p>\n<div id=\"fs-id1167793590440\" class=\"equation unnumbered\">[latex]\\text{ln}9-2\\text{ln}3+\\text{ln}(\\frac{1}{3}).[\/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-id1167794337026\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167794337026\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794337026\">We have<\/p>\n<div id=\"fs-id1167794337029\" class=\"equation unnumbered\">[latex]\\text{ln}9-2\\text{ln}3+\\text{ln}(\\frac{1}{3})=\\text{ln}({3}^{2})-2\\text{ln}3+\\text{ln}({3}^{-1})=2\\text{ln}3-2\\text{ln}3-\\text{ln}3=\\text{\u2212}\\text{ln}3.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793603772\" class=\"textbox exercises checkpoint\">\n<div class=\"exercise\">\n<div id=\"fs-id1167793294690\" class=\"textbox\">\n<p id=\"fs-id1167793607833\">Use properties of logarithms to simplify the following expression into a single logarithm:<\/p>\n<div id=\"fs-id1167793607837\" class=\"equation unnumbered\">[latex]\\text{ln}8-\\text{ln}2-\\text{ln}(\\frac{1}{4}).[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167793949542\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793949542\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793949542\" class=\"hidden-answer\" style=\"display: none\">[latex]4\\text{ln}2[\/latex]<\/div>\n<div id=\"fs-id1167794144762\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1167793949514\">Apply the properties of logarithms.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793514534\" class=\"bc-section section\">\n<h1>Defining the Number [latex]e[\/latex]<\/h1>\n<p id=\"fs-id1167793877994\">Now that we have the natural logarithm defined, we can use that function to define the number [latex]e.[\/latex]<\/p>\n<div id=\"fs-id1167793249163\" class=\"textbox key-takeaways\">\n<div class=\"title\">\n<h3>Definition<\/h3>\n<\/div>\n<p id=\"fs-id1167793249167\">The number [latex]e[\/latex] is defined to be the real number such that<\/p>\n<div id=\"fs-id1167794140148\" class=\"equation unnumbered\">[latex]\\text{ln}e=1.[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1167793871307\">To put it another way, the area under the curve [latex]y=1\\text{\/}t[\/latex] between [latex]t=1[\/latex] and [latex]t=e[\/latex] is 1 (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_06_07_003\">(Figure)<\/a>). The proof that such a number exists and is unique is left to you. (<em>Hint<\/em>: Use the Intermediate Value Theorem to prove existence and the fact that [latex]\\text{ln}x[\/latex] is increasing to prove uniqueness.)<\/p>\n<div id=\"CNX_Calc_Figure_06_07_003\" class=\"wp-caption aligncenter\">\n<div style=\"width: 314px\" 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\/11213325\/CNX_Calc_Figure_06_07_003.jpg\" alt=\"This figure is a graph. It is the curve y=1\/t. It is decreasing and in the first quadrant. Under the curve is a shaded area. The area is bounded to the left at x=1 and to the right at x=e. The area is labeled \u201carea=1\u201d.\" width=\"304\" height=\"316\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. The area under the curve from 1 to [latex]e[\/latex] is equal to one.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793939504\">The number [latex]e[\/latex] can be shown to be irrational, although we won\u2019t do so here (see the Student Project in <a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/HTmjSAcf@2.46:CZi7x6Ls@3\/Taylor-and-Maclaurin-Series\">Taylor and Maclaurin Series<\/a> in the second volume of this text). Its approximate value is given by<\/p>\n<div id=\"fs-id1167793443440\" class=\"equation unnumbered\">[latex]e\\approx 2.71828182846.[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167794051412\" class=\"bc-section section\">\n<h1>The Exponential Function<\/h1>\n<p id=\"fs-id1167794051417\">We now turn our attention to the function [latex]{e}^{x}.[\/latex] Note that the natural logarithm is one-to-one and therefore has an inverse function. For now, we denote this inverse function by [latex]\\text{exp}x.[\/latex] Then,<\/p>\n<div id=\"fs-id1167793514717\" class=\"equation unnumbered\">[latex]\\text{exp}(\\text{ln}x)=x\\text{ for }x>0\\text{ and }\\text{ln}(\\text{exp}x)=x\\text{for all}x.[\/latex]<\/div>\n<p id=\"fs-id1167793932178\">The following figure shows the graphs of [latex]\\text{exp}x[\/latex] and [latex]\\text{ln}x.[\/latex]<\/p>\n<div id=\"CNX_Calc_Figure_06_07_004\" class=\"wp-caption aligncenter\">\n<div style=\"width: 431px\" 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\/11213328\/CNX_Calc_Figure_06_07_004.jpg\" alt=\"This figure is a graph. It has three curves. The first curve is labeled exp x. It is an increasing curve with the x-axis as a horizontal asymptote. It intersects the y-axis at y=1. The second curve is a diagonal line through the origin. The third curve is labeled lnx. It is an increasing curve with the y-axis as an vertical axis. It intersects the x-axis at x=1.\" width=\"421\" height=\"422\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4. The graphs of [latex]\\text{ln}x[\/latex] and [latex]\\text{exp}x.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793905821\">We hypothesize that [latex]\\text{exp}x={e}^{x}.[\/latex] For rational values of [latex]x,[\/latex] this is easy to show. If [latex]x[\/latex] is rational, then we have [latex]\\text{ln}({e}^{x})=x\\text{ln}e=x.[\/latex] Thus, when [latex]x[\/latex] is rational, [latex]{e}^{x}=\\text{exp}x.[\/latex] For irrational values of [latex]x,[\/latex] we simply define [latex]{e}^{x}[\/latex] as the inverse function of [latex]\\text{ln}x.[\/latex]<\/p>\n<div id=\"fs-id1167793579578\" class=\"textbox key-takeaways\">\n<div class=\"title\">\n<h3>Definition<\/h3>\n<\/div>\n<p id=\"fs-id1167793367959\">For any real number [latex]x,[\/latex] define [latex]y={e}^{x}[\/latex] to be the number for which<\/p>\n<div id=\"fs-id1167793447883\" class=\"equation\">[latex]\\text{ln}y=\\text{ln}({e}^{x})=x.[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1167793489536\">Then we have [latex]{e}^{x}=\\text{exp}(x)[\/latex] for all [latex]x,[\/latex] and thus<\/p>\n<div id=\"fs-id1167793564127\" class=\"equation\">[latex]{e}^{\\text{ln}x}=x\\text{ for }x>0\\text{ and }\\text{ln}({e}^{x})=x[\/latex]<\/div>\n<p id=\"fs-id1167794036848\">for all [latex]x.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167793964114\" class=\"bc-section section\">\n<h1>Properties of the Exponential Function<\/h1>\n<p id=\"fs-id1167793516179\">Since the exponential function was defined in terms of an inverse function, and not in terms of a power of [latex]e,[\/latex] we must verify that the usual laws of exponents hold for the function [latex]{e}^{x}.[\/latex]<\/p>\n<div id=\"fs-id1167794176164\" class=\"textbox key-takeaways theorem\">\n<h3>Properties of the Exponential Function<\/h3>\n<p id=\"fs-id1167794210866\">If [latex]p[\/latex] and [latex]q[\/latex] are any real numbers and [latex]r[\/latex] is a rational number, then<\/p>\n<ol id=\"fs-id1167793361174\">\n<li>[latex]{e}^{p}{e}^{q}={e}^{p+q}[\/latex]<\/li>\n<li>[latex]\\frac{{e}^{p}}{{e}^{q}}={e}^{p-q}[\/latex]<\/li>\n<li>[latex]{({e}^{p})}^{r}={e}^{pr}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1167794064034\" class=\"bc-section section\">\n<h2>Proof<\/h2>\n<p id=\"fs-id1167793420459\">Note that if [latex]p[\/latex] and [latex]q[\/latex] are rational, the properties hold. However, if [latex]p[\/latex] or [latex]q[\/latex] are irrational, we must apply the inverse function definition of [latex]{e}^{x}[\/latex] and verify the properties. Only the first property is verified here; the other two are left to you. We have<\/p>\n<div id=\"fs-id1167793301084\" class=\"equation unnumbered\">[latex]\\text{ln}({e}^{p}{e}^{q})=\\text{ln}({e}^{p})+\\text{ln}({e}^{q})=p+q=\\text{ln}({e}^{p+q}).[\/latex]<\/div>\n<p id=\"fs-id1167793937142\">Since [latex]\\text{ln}x[\/latex] is one-to-one, then<\/p>\n<div id=\"fs-id1167794329135\" class=\"equation unnumbered\">[latex]{e}^{p}{e}^{q}={e}^{p+q}.[\/latex]<\/div>\n<p id=\"fs-id1167793503220\">\u25a1<\/p>\n<p id=\"fs-id1167793443585\">As with part iv. of the logarithm properties, we can extend property iii. to irrational values of [latex]r,[\/latex] and we do so by the end of the section.<\/p>\n<p id=\"fs-id1167793952464\">We also want to verify the differentiation formula for the function [latex]y={e}^{x}.[\/latex] To do this, we need to use implicit differentiation. Let [latex]y={e}^{x}.[\/latex] Then<\/p>\n<div id=\"fs-id1167794095942\" class=\"equation unnumbered\">[latex]\\begin{array}{ccc}\\hfill \\text{ln}y& =\\hfill & x\\hfill \\\\ \\hfill \\frac{d}{dx}\\text{ln}y& =\\hfill & \\frac{d}{dx}x\\hfill \\\\ \\hfill \\frac{1}{y}\\frac{dy}{dx}& =\\hfill & 1\\hfill \\\\ \\hfill \\frac{dy}{dx}& =\\hfill & y.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167794147337\">Thus, we see<\/p>\n<div class=\"equation unnumbered\">[latex]\\frac{d}{dx}{e}^{x}={e}^{x}[\/latex]<\/div>\n<p id=\"fs-id1167793370445\">as desired, which leads immediately to the integration formula<\/p>\n<div id=\"fs-id1167793924216\" class=\"equation unnumbered\">[latex]\\int {e}^{x}dx={e}^{x}+C.[\/latex]<\/div>\n<p id=\"fs-id1167793272155\">We apply these formulas in the following examples.<\/p>\n<div id=\"fs-id1167793272158\" class=\"textbox examples\">\n<h3>Using Properties of Exponential Functions<\/h3>\n<div id=\"fs-id1167794098635\" class=\"exercise\">\n<div id=\"fs-id1167794098637\" class=\"textbox\">\n<p id=\"fs-id1167793393570\">Evaluate the following derivatives:<\/p>\n<ol id=\"fs-id1167793393574\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dt}{e}^{3t}{e}^{{t}^{2}}[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{e}^{3{x}^{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-id1167794293256\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167794293256\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794293256\">We apply the chain rule as necessary.<\/p>\n<ol id=\"fs-id1167794293259\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dt}{e}^{3t}{e}^{{t}^{2}}=\\frac{d}{dt}{e}^{3t+{t}^{2}}={e}^{3t+{t}^{2}}(3+2t)[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{e}^{3{x}^{2}}={e}^{3{x}^{2}}6x[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793372329\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1167793442905\" class=\"exercise\">\n<div id=\"fs-id1167793442908\" class=\"textbox\">\n<p id=\"fs-id1167793374990\">Evaluate the following derivatives:<\/p>\n<ol id=\"fs-id1167793374993\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dx}(\\frac{{e}^{{x}^{2}}}{{e}^{5x}})[\/latex]<\/li>\n<li>[latex]\\frac{d}{dt}{({e}^{2t})}^{3}[\/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-id1167793872351\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793872351\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1167793872351\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dx}(\\frac{{e}^{{x}^{2}}}{{e}^{5x}})={e}^{{x}^{2}-5x}(2x-5)[\/latex]<\/li>\n<li>[latex]\\frac{d}{dt}{({e}^{2t})}^{3}=6{e}^{6t}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1167794136708\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1167793568428\">Use the properties of exponential functions and the chain rule as necessary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793944672\" class=\"textbox examples\">\n<h3>Using Properties of Exponential Functions<\/h3>\n<div id=\"fs-id1167794031059\" class=\"exercise\">\n<div id=\"fs-id1167794031061\" class=\"textbox\">\n<p id=\"fs-id1167794020845\">Evaluate the following integral: [latex]\\int 2x{e}^{\\text{\u2212}{x}^{2}}dx.[\/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-id1167793501968\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793501968\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793501968\">Using [latex]u[\/latex]-substitution, let [latex]u=\\text{\u2212}{x}^{2}.[\/latex] Then [latex]du=-2xdx,[\/latex] and we have<\/p>\n<div id=\"fs-id1167793956578\" class=\"equation unnumbered\">[latex]\\int 2x{e}^{\\text{\u2212}{x}^{2}}dx=\\text{\u2212}\\int {e}^{u}du=\\text{\u2212}{e}^{u}+C=\\text{\u2212}{e}^{\\text{\u2212}{x}^{2}}+C.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793478795\" class=\"textbox exercises checkpoint\">\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p id=\"fs-id1167793553724\">Evaluate the following integral: [latex]\\int \\frac{4}{{e}^{3x}}dx.[\/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-id1167793510888\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793510888\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793510888\">[latex]\\int \\frac{4}{{e}^{3x}}dx=-\\frac{4}{3}{e}^{-3x}+C[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167794178061\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1167793943946\">Use the properties of exponential functions and [latex]u\\text{-substitution}[\/latex] as necessary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793400840\" class=\"bc-section section\">\n<h1>General Logarithmic and Exponential Functions<\/h1>\n<p id=\"fs-id1167793400845\">We close this section by looking at exponential functions and logarithms with bases other than [latex]e.[\/latex] Exponential functions are functions of the form [latex]f(x)={a}^{x}.[\/latex] Note that unless [latex]a=e,[\/latex] we still do not have a mathematically rigorous definition of these functions for irrational exponents. Let\u2019s rectify that here by defining the function [latex]f(x)={a}^{x}[\/latex] in terms of the exponential function [latex]{e}^{x}.[\/latex] We then examine logarithms with bases other than [latex]e[\/latex] as inverse functions of exponential functions.<\/p>\n<div id=\"fs-id1167793285142\" class=\"textbox key-takeaways\">\n<div class=\"title\">\n<h3>Definition<\/h3>\n<\/div>\n<p>For any [latex]a>0,[\/latex] and for any real number [latex]x,[\/latex] define [latex]y={a}^{x}[\/latex] as follows:<\/p>\n<div id=\"fs-id1167793420642\" class=\"equation unnumbered\">[latex]y={a}^{x}={e}^{x\\text{ln}a}.[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1167793455076\">Now [latex]{a}^{x}[\/latex] is defined rigorously for all values of [latex]x[\/latex]. This definition also allows us to generalize property iv. of logarithms and property iii. of exponential functions to apply to both rational and irrational values of [latex]r.[\/latex] It is straightforward to show that properties of exponents hold for general exponential functions defined in this way.<\/p>\n<p id=\"fs-id1167793384514\">Let\u2019s now apply this definition to calculate a differentiation formula for [latex]{a}^{x}.[\/latex] We have<\/p>\n<div id=\"fs-id1167793559160\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}{a}^{x}=\\frac{d}{dx}{e}^{x\\text{ln}a}={e}^{x\\text{ln}a}\\text{ln}a={a}^{x}\\text{ln}a.[\/latex]<\/div>\n<p id=\"fs-id1167794075642\">The corresponding integration formula follows immediately.<\/p>\n<div id=\"fs-id1167794075645\" class=\"textbox key-takeaways theorem\">\n<h3>Derivatives and Integrals Involving General Exponential Functions<\/h3>\n<p id=\"fs-id1167793293670\">Let [latex]a>0.[\/latex] Then,<\/p>\n<div id=\"fs-id1167793271586\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}{a}^{x}={a}^{x}\\text{ln}a[\/latex]<\/div>\n<p id=\"fs-id1167793562027\">and<\/p>\n<div id=\"fs-id1167793562030\" class=\"equation unnumbered\">[latex]\\int {a}^{x}dx=\\frac{1}{\\text{ln}a}{a}^{x}+C.[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1167793267617\">If [latex]a\\ne 1,[\/latex] then the function [latex]{a}^{x}[\/latex] is one-to-one and has a well-defined inverse. Its inverse is denoted by [latex]{\\text{log}}_{a}x.[\/latex] Then,<\/p>\n<div id=\"fs-id1167793776857\" class=\"equation unnumbered\">[latex]y={\\text{log}}_{a}x\\text{if and only if}x={a}^{y}.[\/latex]<\/div>\n<p id=\"fs-id1167793929151\">Note that general logarithm functions can be written in terms of the natural logarithm. Let [latex]y={\\text{log}}_{a}x.[\/latex] Then, [latex]x={a}^{y}.[\/latex] Taking the natural logarithm of both sides of this second equation, we get<\/p>\n<div class=\"equation unnumbered\">[latex]\\begin{array}{ccc}\\hfill \\text{ln}x& =\\hfill & \\text{ln}({a}^{y})\\hfill \\\\ \\hfill \\text{ln}x& =\\hfill & y\\text{ln}a\\hfill \\\\ \\hfill y& =\\hfill & \\frac{\\text{ln}x}{\\text{ln}a}\\hfill \\\\ \\hfill {\\text{log}}_{}x& =\\hfill & \\frac{\\text{ln}x}{\\text{ln}a}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167793450615\">Thus, we see that all logarithmic functions are constant multiples of one another. Next, we use this formula to find a differentiation formula for a logarithm with base [latex]a.[\/latex] Again, let [latex]y={\\text{log}}_{a}x.[\/latex] Then,<\/p>\n<div class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill \\frac{dy}{dx}& =\\frac{d}{dx}({\\text{log}}_{a}x)\\hfill \\\\ & =\\frac{d}{dx}(\\frac{\\text{ln}x}{\\text{ln}a})\\hfill \\\\ & =(\\frac{1}{\\text{ln}a})\\frac{d}{dx}(\\text{ln}x)\\hfill \\\\ & =\\frac{1}{\\text{ln}a}\u00b7\\frac{1}{x}\\hfill \\\\ & =\\frac{1}{x\\text{ln}a}.\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1167794324570\" class=\"textbox key-takeaways theorem\">\n<h3>Derivatives of General Logarithm Functions<\/h3>\n<p id=\"fs-id1167793432748\">Let [latex]a>0.[\/latex] Then,<\/p>\n<div id=\"fs-id1167794139845\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}{\\text{log}}_{a}x=\\frac{1}{x\\text{ln}a}.[\/latex]<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1167793948186\" class=\"exercise\">\n<div id=\"fs-id1167793948188\" class=\"textbox\">\n<h3>Calculating Derivatives of General Exponential and Logarithm Functions<\/h3>\n<p id=\"fs-id1167793640049\">Evaluate the following derivatives:<\/p>\n<ol id=\"fs-id1167793640052\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dt}({4}^{t}\u00b7{2}^{{t}^{2}})[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{\\text{log}}_{8}(7{x}^{2}+4)[\/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-id1167793298214\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793298214\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793298214\">We need to apply the chain rule as necessary.<\/p>\n<ol id=\"fs-id1167793829825\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dt}({4}^{t}\u00b7{2}^{{t}^{2}})=\\frac{d}{dt}({2}^{2t}\u00b7{2}^{{t}^{2}})=\\frac{d}{dt}({2}^{2t+{t}^{2}})={2}^{2t+{t}^{2}}\\text{ln}(2)(2+2t)[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{\\text{log}}_{8}(7{x}^{2}+4)=\\frac{1}{(7{x}^{2}+4)(\\text{ln}8)}(14x)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793455288\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1167793455291\" class=\"exercise\">\n<div id=\"fs-id1167793455293\" class=\"textbox\">\n<p id=\"fs-id1167793543534\">Evaluate the following derivatives:<\/p>\n<ol style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dt}{4}^{{t}^{4}}[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{\\text{log}}_{3}(\\sqrt{{x}^{2}+1})[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1167793978416\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793978416\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793978416\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dt}{4}^{{t}^{4}}={4}^{{t}^{4}}(\\text{ln}4)(4{t}^{3})[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{\\text{log}}_{3}(\\sqrt{{x}^{2}+1})=\\frac{x}{(\\text{ln}3)({x}^{2}+1)}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1167793521316\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p>Use the formulas and apply the chain rule as necessary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793499138\" class=\"textbox examples\">\n<div class=\"exercise\">\n<div class=\"textbox\">\n<h3>Integrating General Exponential Functions<\/h3>\n<p id=\"fs-id1167793499148\">Evaluate the following integral: [latex]\\int \\frac{3}{{2}^{3x}}dx.[\/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-id1167793956537\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793956537\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793956537\">Use [latex]u\\text{-substitution}[\/latex] and let [latex]u=-3x.[\/latex] Then [latex]du=-3dx[\/latex] and we have<\/p>\n<div id=\"fs-id1167793929418\" class=\"equation unnumbered\">[latex]\\int \\frac{3}{{2}^{3x}}dx=\\int 3\u00b7{2}^{-3x}dx=\\text{\u2212}\\int {2}^{u}du=-\\frac{1}{\\text{ln}2}{2}^{u}+C=-\\frac{1}{\\text{ln}2}{2}^{-3x}+C.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793789604\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1167793789608\" class=\"exercise\">\n<div id=\"fs-id1167794188287\" class=\"textbox\">\n<p id=\"fs-id1167794188289\">Evaluate the following integral: [latex]\\int {x}^{2}{2}^{{x}^{3}}dx.[\/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-id1167793979123\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793979123\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793979123\">[latex]\\int {x}^{2}{2}^{{x}^{3}}dx=\\frac{1}{3\\text{ln}2}{2}^{{x}^{3}}+C[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167793638224\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1167793423554\">Use the properties of exponential functions and [latex]u\\text{-substitution}[\/latex] as necessary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793423569\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1167793931233\">\n<li>The earlier treatment of logarithms and exponential functions did not define the functions precisely and formally. This section develops the concepts in a mathematically rigorous way.<\/li>\n<li>The cornerstone of the development is the definition of the natural logarithm in terms of an integral.<\/li>\n<li>The function [latex]{e}^{x}[\/latex] is then defined as the inverse of the natural logarithm.<\/li>\n<li>General exponential functions are defined in terms of [latex]{e}^{x},[\/latex] and the corresponding inverse functions are general logarithms.<\/li>\n<li>Familiar properties of logarithms and exponents still hold in this more rigorous context.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1167794327228\" class=\"key-equations\">\n<h1>Key Equations<\/h1>\n<ul id=\"fs-id1167793278406\">\n<li><strong>Natural logarithm function<\/strong><\/li>\n<li>[latex]\\text{ln}x={\\int }_{1}^{x}\\frac{1}{t}dt[\/latex] Z<\/li>\n<li><strong>Exponential function<\/strong>[latex]y={e}^{x}[\/latex]<\/li>\n<li>[latex]\\text{ln}y=\\text{ln}({e}^{x})=x[\/latex] Z<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1167793879358\" class=\"textbox exercises\">\n<p id=\"fs-id1167793879362\">For the following exercises, find the derivative [latex]\\frac{dy}{dx}.[\/latex]<\/p>\n<div id=\"fs-id1167793363071\" class=\"exercise\">\n<div id=\"fs-id1167793363073\" class=\"textbox\">\n<p id=\"fs-id1167793363076\">[latex]y=\\text{ln}(2x)[\/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-id1167794097576\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167794097576\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794097576\">[latex]\\frac{1}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794097588\" class=\"exercise\">\n<div id=\"fs-id1167794226012\" class=\"textbox\">\n<p id=\"fs-id1167794226014\">[latex]y=\\text{ln}(2x+1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793831549\" class=\"exercise\">\n<div id=\"fs-id1167793629438\" class=\"textbox\">\n<p id=\"fs-id1167793629440\">[latex]y=\\frac{1}{\\text{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-id1167793546866\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793546866\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793546866\">[latex]-\\frac{1}{x{(\\text{ln}x)}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793541105\">For the following exercises, find the indefinite integral.<\/p>\n<div id=\"fs-id1167793541108\" class=\"exercise\">\n<div id=\"fs-id1167793541110\" class=\"textbox\">\n<p id=\"fs-id1167793541112\">[latex]\\int \\frac{dt}{3t}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793367825\" class=\"exercise\">\n<div id=\"fs-id1167793367827\" class=\"textbox\">\n<p id=\"fs-id1167793367829\">[latex]\\int \\frac{dx}{1+x}[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p>[latex]\\text{ln}(x+1)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793706099\">For the following exercises, find the derivative [latex]dy\\text{\/}dx.[\/latex] (You can use a calculator to plot the function and the derivative to confirm that it is correct.)<\/p>\n<div id=\"fs-id1167793706119\" class=\"exercise\">\n<div id=\"fs-id1167793706121\" class=\"textbox\">\n<p id=\"fs-id1167793706123\"><strong>[T]<\/strong>[latex]y=\\frac{\\text{ln}(x)}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793510876\" class=\"exercise\">\n<div id=\"fs-id1167793510878\" class=\"textbox\">\n<p id=\"fs-id1167793287409\"><strong>[T]<\/strong>[latex]y=x\\text{ln}(x)[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p id=\"fs-id1167793628620\">[latex]\\text{ln}(x)+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793628640\" class=\"exercise\">\n<div id=\"fs-id1167793628642\" class=\"textbox\">\n<p id=\"fs-id1167793628644\"><strong>[T]<\/strong>[latex]y={\\text{log}}_{10}x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793244348\" class=\"exercise\">\n<div id=\"fs-id1167793244350\" class=\"textbox\">\n<p id=\"fs-id1167793244352\"><strong>[T]<\/strong>[latex]y=\\text{ln}( \\sin 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-id1167793776894\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793776894\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793776894\">[latex]\\cot (x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793504037\" class=\"exercise\">\n<div class=\"textbox\">\n<p id=\"fs-id1167793504041\"><strong>[T]<\/strong>[latex]y=\\text{ln}(\\text{ln}x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793423443\" class=\"exercise\">\n<div id=\"fs-id1167793423445\" class=\"textbox\">\n<p id=\"fs-id1167793423447\"><strong>[T]<\/strong>[latex]y=7\\text{ln}(4x)[\/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-id1167793293693\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793293693\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793293693\">[latex]\\frac{7}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793293705\" class=\"exercise\">\n<div id=\"fs-id1167793293707\" class=\"textbox\">\n<p id=\"fs-id1167793293709\"><strong>[T]<\/strong>[latex]y=\\text{ln}({(4x)}^{7})[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794073092\" class=\"exercise\">\n<div id=\"fs-id1167794073094\" class=\"textbox\">\n<p id=\"fs-id1167794073096\"><strong>[T]<\/strong>[latex]y=\\text{ln}( \\tan 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-id1167793557826\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793557826\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793557826\">[latex]\\csc (x) \\sec x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793563636\" class=\"exercise\">\n<div id=\"fs-id1167793563639\" class=\"textbox\">\n<p id=\"fs-id1167793563641\"><strong>[T]<\/strong>[latex]y=\\text{ln}( \\tan (3x))[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id1167793281524\" class=\"textbox\">\n<p id=\"fs-id1167793281526\"><strong>[T]<\/strong>[latex]y=\\text{ln}({ \\cos }^{2}x)[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p>[latex]-2 \\tan x[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793553757\">For the following exercises, find the definite or indefinite integral.<\/p>\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p>[latex]{\\int }_{0}^{1}\\frac{dx}{3+x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793372398\" class=\"exercise\">\n<div id=\"fs-id1167793372400\" class=\"textbox\">\n<p id=\"fs-id1167793372402\">[latex]{\\int }_{0}^{1}\\frac{dt}{3+2t}[\/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-id1167793553651\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793553651\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793553651\">[latex]\\frac{1}{2}\\text{ln}(\\frac{5}{3})[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793582462\" class=\"exercise\">\n<div id=\"fs-id1167793582464\" class=\"textbox\">\n<p id=\"fs-id1167793582466\">[latex]{\\int }_{0}^{2}\\frac{xdx}{{x}^{2}+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794122084\" class=\"exercise\">\n<div id=\"fs-id1167794122086\" class=\"textbox\">\n<p id=\"fs-id1167794122088\">[latex]{\\int }_{0}^{2}\\frac{{x}^{3}dx}{{x}^{2}+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-id1167793952145\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793952145\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793952145\">[latex]2-\\frac{1}{2}\\text{ln}(5)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793499058\" class=\"exercise\">\n<div id=\"fs-id1167793499060\" class=\"textbox\">\n<p id=\"fs-id1167793499062\">[latex]{\\int }_{2}^{e}\\frac{dx}{x\\text{ln}x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794054231\" class=\"exercise\">\n<div id=\"fs-id1167794054233\" class=\"textbox\">\n<p id=\"fs-id1167794054235\">[latex]{\\int }_{2}^{e}\\frac{dx}{{(x\\text{ln}(x))}^{2}}[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p id=\"fs-id1167793473569\">[latex]\\frac{1}{\\text{ln}(2)}-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793481974\" class=\"exercise\">\n<div id=\"fs-id1167793481976\" class=\"textbox\">\n<p id=\"fs-id1167793481978\">[latex]\\int \\frac{ \\cos xdx}{ \\sin x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793469786\" class=\"exercise\">\n<div id=\"fs-id1167793469788\" class=\"textbox\">\n<p id=\"fs-id1167793469791\">[latex]{\\int }_{0}^{\\pi \\text{\/}4} \\tan xdx[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167794005215\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167794005215\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167794005215\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{1}{2}\\text{ln}(2)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794005239\" class=\"exercise\">\n<div id=\"fs-id1167794005241\" class=\"textbox\">\n<p id=\"fs-id1167794005243\">[latex]\\int \\cot (3x)dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794291581\" class=\"exercise\">\n<div id=\"fs-id1167794291583\" class=\"textbox\">\n<p id=\"fs-id1167794291585\">[latex]\\int \\frac{{(\\text{ln}x)}^{2}dx}{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-id1167793421199\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793421199\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793421199\">[latex]\\frac{1}{3}{(\\text{ln}x)}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>For the following exercises, compute [latex]dy\\text{\/}dx[\/latex] by differentiating [latex]\\text{ln}y.[\/latex]<\/p>\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p>[latex]y=\\sqrt{{x}^{2}+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793394990\" class=\"exercise\">\n<div id=\"fs-id1167793394992\" class=\"textbox\">\n<p id=\"fs-id1167793394994\">[latex]y=\\sqrt{{x}^{2}+1}\\sqrt{{x}^{2}-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-id1167793595181\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793595181\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793595181\">[latex]\\frac{2{x}^{3}}{\\sqrt{{x}^{2}+1}\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793595222\" class=\"exercise\">\n<div id=\"fs-id1167793595224\" class=\"textbox\">\n<p id=\"fs-id1167793595226\">[latex]y={e}^{ \\sin x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793634250\" class=\"exercise\">\n<div id=\"fs-id1167793634252\" class=\"textbox\">\n<p id=\"fs-id1167793634254\">[latex]y={x}^{-1\\text{\/}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-id1167793465231\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793465231\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793465231\">[latex]{x}^{-2-(1\\text{\/}x)}(\\text{ln}x-1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793465279\" class=\"exercise\">\n<div id=\"fs-id1167793465281\" class=\"textbox\">\n<p id=\"fs-id1167793465283\">[latex]y={e}^{(ex)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p>[latex]y={x}^{e}[\/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-id1167793445732\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793445732\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793445732\">[latex]e{x}^{e-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793445751\" class=\"exercise\">\n<div id=\"fs-id1167793445753\" class=\"textbox\">\n<p id=\"fs-id1167793445755\">[latex]y={x}^{(ex)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793543331\" class=\"exercise\">\n<div id=\"fs-id1167793543333\" class=\"textbox\">\n<p id=\"fs-id1167793543335\">[latex]y=\\sqrt{x}\\sqrt[3]{x}\\sqrt[6]{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-id1167793566015\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793566015\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793566015\">1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793566023\" class=\"exercise\">\n<div id=\"fs-id1167793566025\" class=\"textbox\">\n<p id=\"fs-id1167793566027\">[latex]y={x}^{-1\\text{\/}\\text{ln}x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793566062\" class=\"exercise\">\n<div id=\"fs-id1167793566064\" class=\"textbox\">\n<p id=\"fs-id1167793566066\">[latex]y={e}^{\\text{\u2212}\\text{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-id1167793315539\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793315539\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793315539\">[latex]-\\frac{1}{{x}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793315557\">For the following exercises, evaluate by any method.<\/p>\n<div id=\"fs-id1167793315561\" class=\"exercise\">\n<div id=\"fs-id1167793315563\" class=\"textbox\">\n<p id=\"fs-id1167793315565\">[latex]{\\int }_{5}^{10}\\frac{dt}{t}-{\\int }_{5x}^{10x}\\frac{dt}{t}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793937978\" class=\"exercise\">\n<div class=\"textbox\">\n<p>[latex]{\\int }_{1}^{{e}^{\\pi }}\\frac{dx}{x}+{\\int }_{-2}^{-1}\\frac{dx}{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-id1167793455346\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793455346\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793455346\">[latex]\\pi -\\text{ln}(2)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id1167794127470\" class=\"textbox\">\n<p id=\"fs-id1167794127472\">[latex]\\frac{d}{dx}{\\int }_{x}^{1}\\frac{dt}{t}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794127523\" class=\"exercise\">\n<div id=\"fs-id1167794127525\" class=\"textbox\">\n<p id=\"fs-id1167794127527\">[latex]\\frac{d}{dx}{\\int }_{x}^{{x}^{2}}\\frac{dt}{t}[\/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-id1167794146824\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167794146824\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794146824\">[latex]\\frac{1}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794146835\" class=\"exercise\">\n<div id=\"fs-id1167794146838\" class=\"textbox\">\n<p id=\"fs-id1167794146840\">[latex]\\frac{d}{dx}\\text{ln}( \\sec x+ \\tan x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793423282\">For the following exercises, use the function [latex]\\text{ln}x.[\/latex] If you are unable to find intersection points analytically, use a calculator.<\/p>\n<div id=\"fs-id1167793423297\" class=\"exercise\">\n<div id=\"fs-id1167793423299\" class=\"textbox\">\n<p id=\"fs-id1167793423301\">Find the area of the region enclosed by [latex]x=1[\/latex] and [latex]y=5[\/latex] above [latex]y=\\text{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-id1167793541846\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793541846\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793541846\">[latex]{e}^{5}-6{\\text{units}}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793541870\" class=\"exercise\">\n<div id=\"fs-id1167793541872\" class=\"textbox\">\n<p id=\"fs-id1167793541874\"><strong>[T]<\/strong> Find the arc length of [latex]\\text{ln}x[\/latex] from [latex]x=1[\/latex] to [latex]x=2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793454016\" class=\"exercise\">\n<div id=\"fs-id1167793454018\" class=\"textbox\">\n<p id=\"fs-id1167793454020\">Find the area between [latex]\\text{ln}x[\/latex] and the [latex]x[\/latex]-axis from [latex]x=1\\text{ to }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-id1167793713050\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793713050\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793713050\">[latex]\\text{ln}(4)-1{\\text{units}}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793713080\" class=\"exercise\">\n<div id=\"fs-id1167793713082\" class=\"textbox\">\n<p id=\"fs-id1167793713084\">Find the volume of the shape created when rotating this curve from [latex]x=1\\text{ to }x=2[\/latex] around the [latex]x[\/latex]-axis, as pictured here.<\/p>\n<p><span id=\"fs-id1167793960058\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213332\/CNX_Calc_Figure_06_07_201.jpg\" alt=\"This figure is a surface. It has been generated by revolving the curve ln x about the x-axis. The surface is inside of a cube showing it is 3-dimensinal.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793510646\" class=\"exercise\">\n<div id=\"fs-id1167793510649\" class=\"textbox\">\n<p id=\"fs-id1167793510651\"><strong>[T]<\/strong> Find the surface area of the shape created when rotating the curve in the previous exercise from [latex]x=1[\/latex] to [latex]x=2[\/latex] around the [latex]x[\/latex]-axis.<\/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-id1167793510687\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793510687\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793510687\">2.8656<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793510695\">If you are unable to find intersection points analytically in the following exercises, use a calculator.<\/p>\n<div id=\"fs-id1167793510699\" class=\"exercise\">\n<div id=\"fs-id1167793510702\" class=\"textbox\">\n<p id=\"fs-id1167793510704\">Find the area of the hyperbolic quarter-circle enclosed by [latex]x=2\\text{ and }y=2[\/latex] above [latex]y=1\\text{\/}x.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794209645\" class=\"exercise\">\n<div id=\"fs-id1167794209648\" class=\"textbox\">\n<p id=\"fs-id1167794209650\"><strong>[T]<\/strong> Find the arc length of [latex]y=1\\text{\/}x[\/latex] from [latex]x=1\\text{ to }x=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-id1167793570742\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793570742\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793570742\">3.1502<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793570750\" class=\"exercise\">\n<div id=\"fs-id1167793570752\" class=\"textbox\">\n<p id=\"fs-id1167793570755\">Find the area under [latex]y=1\\text{\/}x[\/latex] and above the [latex]x[\/latex]-axis from [latex]x=1\\text{ to }x=4.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793593495\">For the following exercises, verify the derivatives and antiderivatives.<\/p>\n<div id=\"fs-id1167793593498\" class=\"exercise\">\n<div id=\"fs-id1167793593500\" class=\"textbox\">\n<p id=\"fs-id1167793593502\">[latex]\\frac{d}{dx}\\text{ln}(x+\\sqrt{{x}^{2}+1})=\\frac{1}{\\sqrt{1+{x}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793377930\" class=\"exercise\">\n<div id=\"fs-id1167793377932\" class=\"textbox\">\n<p id=\"fs-id1167793377934\">[latex]\\frac{d}{dx}\\text{ln}(\\frac{x-a}{x+a})=\\frac{2a}{({x}^{2}-{a}^{2})}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793380054\" class=\"exercise\">\n<div id=\"fs-id1167793380056\" class=\"textbox\">\n<p id=\"fs-id1167793380058\">[latex]\\frac{d}{dx}\\text{ln}(\\frac{1+\\sqrt{1-{x}^{2}}}{x})=-\\frac{1}{x\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793911914\" class=\"exercise\">\n<div id=\"fs-id1167793911916\" class=\"textbox\">\n<p id=\"fs-id1167793911918\">[latex]\\frac{d}{dx}\\text{ln}(x+\\sqrt{{x}^{2}-{a}^{2}})=\\frac{1}{\\sqrt{{x}^{2}-{a}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793373565\" class=\"exercise\">\n<div id=\"fs-id1167793373567\" class=\"textbox\">\n<p id=\"fs-id1167793373569\">[latex]\\int \\frac{dx}{x\\text{ln}(x)\\text{ln}(\\text{ln}x)}=\\text{ln}(\\text{ln}(\\text{ln}x))+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":311,"menu_order":8,"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-2196","chapter","type-chapter","status-publish","hentry"],"part":2032,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/2196","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":5,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/2196\/revisions"}],"predecessor-version":[{"id":2577,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/2196\/revisions\/2577"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/parts\/2032"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/2196\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/media?parent=2196"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapter-type?post=2196"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/contributor?post=2196"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/license?post=2196"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}