{"id":165,"date":"2023-09-20T22:47:59","date_gmt":"2023-09-20T22:47:59","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/exponential-functions\/"},"modified":"2024-08-04T12:17:05","modified_gmt":"2024-08-04T12:17:05","slug":"exponential-and-logarithmic-functions-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/exponential-and-logarithmic-functions-learn-it-1\/","title":{"raw":"Exponential and Logarithmic Functions: Learn It 1","rendered":"Exponential and Logarithmic Functions: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Work with exponential functions to find their values<\/li>\r\n\t<li>Recognize logarithmic functions, explore their relationship with exponential functions, and change their bases<\/li>\r\n\t<li>Identify hyperbolic functions their graphs, and understand their fundamental identities<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Exponential Functions<\/h2>\r\n<p id=\"fs-id1170572246259\">Exponential functions arise in many applications. One common example is <span class=\"no-emphasis\">population growth<\/span>.<\/p>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572449480\">If a population starts with [latex]P_0[\/latex] individuals and then grows at an annual rate of [latex]2\\%[\/latex], its population after [latex]1[\/latex] year is<\/p>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]P(1)=P_0+0.02P_0=P_0(1+0.02)=P_0(1.02)[\/latex]<\/div>\r\n<p id=\"fs-id1170572092410\">Its population after [latex]2[\/latex] years is<\/p>\r\n<div id=\"fs-id1170572177937\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]P(2)=P(1)+0.02P(1)=P(1)(1.02)=P_0(1.02)^2[\/latex]<\/div>\r\n<p id=\"fs-id1170572130048\">In general, its population after [latex]t[\/latex] years is<\/p>\r\n<div id=\"fs-id1170572280288\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]P(t)=P_0(1.02)^t[\/latex],<\/div>\r\n<p>which is an exponential function.<\/p>\r\n<\/section>\r\n<p>More generally, any function of the form [latex]f(x)=b^x[\/latex], where [latex]b&gt;0, \\, b \\ne 1[\/latex], is an <strong>exponential function<\/strong> with base [latex]b[\/latex] and exponent [latex]x[\/latex]. Exponential functions have constant bases and variable exponents.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>exponential function<\/h3>\r\n\r\nFor any real number [latex]x[\/latex], an exponential function is a function with the form\r\n\r\n<p style=\"text-align: center;\">[latex]f(x)=ab^x[\/latex]<\/p>\r\n\r\nwhere,\r\n\r\n<ul>\r\n\t<li>[latex]a[\/latex] is a non-zero real number called the initial value and<\/li>\r\n\t<li>[latex]b[\/latex] is any positive real number ([latex]b&gt;0[\/latex]) such that [latex]b\u22601[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox questionHelp\">\r\n<p>Why do we limit the base [latex]b[\/latex] to positive values?<\/p>\r\n<p>To ensure that the outputs will be real numbers. Observe what happens if the base is not positive:<\/p>\r\n<ul>\r\n\t<li>Let [latex]b=\u22129[\/latex] and [latex]x=\\frac{1}{2}[\/latex]. Then [latex]f(x)=f(\\frac{1}{2})=(\u22129)^\\frac{1}{2}=\\sqrt{\u22129}[\/latex], which is not a real number.<\/li>\r\n<\/ul>\r\n<p>Why do we limit the base to positive values other than [latex]1[\/latex]?<\/p>\r\n<p>Because base [latex]1[\/latex] results in the constant function. Observe what happens if the base is [latex]1[\/latex]:<\/p>\r\n<ul>\r\n\t<li>Let [latex]b=1[\/latex]. Then [latex]f(x)=1^x=1[\/latex] for any value of [latex]x[\/latex].<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p>Note that a function of the form [latex]f(x)=x^b[\/latex] for some constant [latex]b[\/latex] is not an exponential function but a power function.<\/p>\r\n<p id=\"fs-id1170572248051\">To see the difference between an exponential function and a power function, we can compare the functions [latex]y=x^2[\/latex] and [latex]y=2^x[\/latex].<\/p>\r\n<p>In the table below, we see that both [latex]2^x[\/latex] and [latex]x^2[\/latex] approach infinity as [latex]x \\to \\infty[\/latex]. Eventually, however, [latex]2^x[\/latex] becomes larger than [latex]x^2[\/latex] and grows more rapidly as [latex]x \\to \\infty[\/latex]. In the opposite direction, as [latex]x \\to \u2212\\infty, \\, x^2 \\to \\infty[\/latex], whereas [latex]2^x \\to 0[\/latex]. The line [latex]y=0[\/latex] is a horizontal asymptote for [latex]y=2^x[\/latex].<\/p>\r\n<table class=\"center\" style=\"width: 60%;\">\r\n<caption>Values of [latex]x^2[\/latex] and [latex]2^x[\/latex]<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]\\mathbf{x}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]\\mathbf{x^2}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]\\mathbf{2^x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\">[latex]-3[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]9[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]1\/8[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]1\/4[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]1\/2[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\">[latex]3[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]9[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]16[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]16[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\">[latex]5[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]25[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]32[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\">[latex]6[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]36[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]64[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202644\/CNX_Calc_Figure_01_05_001.jpg\" alt=\"An image of a graph. The x axis runs from -10 to 10 and the y axis runs from 0 to 50. The graph is of two functions. The first function is \u201cy = x squared\u201d, which is a parabola. The function decreases until it hits the origin and then begins increasing. The second function is \u201cy = 2 to the power of x\u201d, which starts slightly above the x axis, and begins increasing very rapidly, more rapidly than the first function.\" width=\"325\" height=\"427\" \/> Figure 1. Both [latex]2^x[\/latex] and [latex]x^2[\/latex] approach infinity as [latex]x \\to \\infty[\/latex], but [latex]2^x[\/latex] grows more rapidly than [latex]x^2[\/latex]. As [latex]x \\to \u2212\\infty, \\, x^2 \\to \\infty[\/latex], whereas [latex]2^x \\to 0[\/latex].[\/caption]\r\n<\/section>\r\n<section class=\"textbox recall\">\r\n<table style=\"height: 240px;\">\r\n<thead>\r\n<tr style=\"height: 15px;\">\r\n<th style=\"text-align: center; height: 15px; width: 554.097px;\" colspan=\"2\">Arrow Notation<\/th>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<th style=\"text-align: center; height: 15px; width: 161.875px;\">Symbol<\/th>\r\n<th style=\"text-align: center; height: 15px; width: 380.764px;\">Meaning<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 161.875px;\">[latex]x\\to \\infty[\/latex]<\/td>\r\n<td style=\"height: 30px; width: 380.764px;\">[latex]x[\/latex] approaches infinity ([latex]x[\/latex]\u00a0increases without bound)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 161.875px;\">[latex]x\\to -\\infty [\/latex]<\/td>\r\n<td style=\"height: 30px; width: 380.764px;\">[latex]x[\/latex] approaches negative infinity ([latex]x[\/latex]\u00a0decreases without bound)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 161.875px;\">[latex]f\\left(x\\right)\\to \\infty [\/latex]<\/td>\r\n<td style=\"height: 30px; width: 380.764px;\">the output approaches infinity (the output increases without bound)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 161.875px;\">[latex]f\\left(x\\right)\\to -\\infty [\/latex]<\/td>\r\n<td style=\"height: 30px; width: 380.764px;\">the output approaches negative infinity (the output decreases without bound)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 161.875px;\">[latex]f\\left(x\\right)\\to a[\/latex]<\/td>\r\n<td style=\"height: 30px; width: 380.764px;\">the output approaches [latex]a[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Work with exponential functions to find their values<\/li>\n<li>Recognize logarithmic functions, explore their relationship with exponential functions, and change their bases<\/li>\n<li>Identify hyperbolic functions their graphs, and understand their fundamental identities<\/li>\n<\/ul>\n<\/section>\n<h2>Exponential Functions<\/h2>\n<p id=\"fs-id1170572246259\">Exponential functions arise in many applications. One common example is <span class=\"no-emphasis\">population growth<\/span>.<\/p>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572449480\">If a population starts with [latex]P_0[\/latex] individuals and then grows at an annual rate of [latex]2\\%[\/latex], its population after [latex]1[\/latex] year is<\/p>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]P(1)=P_0+0.02P_0=P_0(1+0.02)=P_0(1.02)[\/latex]<\/div>\n<p id=\"fs-id1170572092410\">Its population after [latex]2[\/latex] years is<\/p>\n<div id=\"fs-id1170572177937\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]P(2)=P(1)+0.02P(1)=P(1)(1.02)=P_0(1.02)^2[\/latex]<\/div>\n<p id=\"fs-id1170572130048\">In general, its population after [latex]t[\/latex] years is<\/p>\n<div id=\"fs-id1170572280288\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]P(t)=P_0(1.02)^t[\/latex],<\/div>\n<p>which is an exponential function.<\/p>\n<\/section>\n<p>More generally, any function of the form [latex]f(x)=b^x[\/latex], where [latex]b>0, \\, b \\ne 1[\/latex], is an <strong>exponential function<\/strong> with base [latex]b[\/latex] and exponent [latex]x[\/latex]. Exponential functions have constant bases and variable exponents.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>exponential function<\/h3>\n<p>For any real number [latex]x[\/latex], an exponential function is a function with the form<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=ab^x[\/latex]<\/p>\n<p>where,<\/p>\n<ul>\n<li>[latex]a[\/latex] is a non-zero real number called the initial value and<\/li>\n<li>[latex]b[\/latex] is any positive real number ([latex]b>0[\/latex]) such that [latex]b\u22601[\/latex].<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p>Why do we limit the base [latex]b[\/latex] to positive values?<\/p>\n<p>To ensure that the outputs will be real numbers. Observe what happens if the base is not positive:<\/p>\n<ul>\n<li>Let [latex]b=\u22129[\/latex] and [latex]x=\\frac{1}{2}[\/latex]. Then [latex]f(x)=f(\\frac{1}{2})=(\u22129)^\\frac{1}{2}=\\sqrt{\u22129}[\/latex], which is not a real number.<\/li>\n<\/ul>\n<p>Why do we limit the base to positive values other than [latex]1[\/latex]?<\/p>\n<p>Because base [latex]1[\/latex] results in the constant function. Observe what happens if the base is [latex]1[\/latex]:<\/p>\n<ul>\n<li>Let [latex]b=1[\/latex]. Then [latex]f(x)=1^x=1[\/latex] for any value of [latex]x[\/latex].<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox proTip\">\n<p>Note that a function of the form [latex]f(x)=x^b[\/latex] for some constant [latex]b[\/latex] is not an exponential function but a power function.<\/p>\n<p id=\"fs-id1170572248051\">To see the difference between an exponential function and a power function, we can compare the functions [latex]y=x^2[\/latex] and [latex]y=2^x[\/latex].<\/p>\n<p>In the table below, we see that both [latex]2^x[\/latex] and [latex]x^2[\/latex] approach infinity as [latex]x \\to \\infty[\/latex]. Eventually, however, [latex]2^x[\/latex] becomes larger than [latex]x^2[\/latex] and grows more rapidly as [latex]x \\to \\infty[\/latex]. In the opposite direction, as [latex]x \\to \u2212\\infty, \\, x^2 \\to \\infty[\/latex], whereas [latex]2^x \\to 0[\/latex]. The line [latex]y=0[\/latex] is a horizontal asymptote for [latex]y=2^x[\/latex].<\/p>\n<table class=\"center\" style=\"width: 60%;\">\n<caption>Values of [latex]x^2[\/latex] and [latex]2^x[\/latex]<\/caption>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">[latex]\\mathbf{x}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\\mathbf{x^2}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\\mathbf{2^x}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]-3[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]9[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]1\/8[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]-2[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]1\/4[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]-1[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]1\/2[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]3[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]9[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]16[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]16[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]5[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]25[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]32[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\">[latex]6[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]36[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]64[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure style=\"width: 325px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202644\/CNX_Calc_Figure_01_05_001.jpg\" alt=\"An image of a graph. The x axis runs from -10 to 10 and the y axis runs from 0 to 50. The graph is of two functions. The first function is \u201cy = x squared\u201d, which is a parabola. The function decreases until it hits the origin and then begins increasing. The second function is \u201cy = 2 to the power of x\u201d, which starts slightly above the x axis, and begins increasing very rapidly, more rapidly than the first function.\" width=\"325\" height=\"427\" \/><figcaption class=\"wp-caption-text\">Figure 1. Both [latex]2^x[\/latex] and [latex]x^2[\/latex] approach infinity as [latex]x \\to \\infty[\/latex], but [latex]2^x[\/latex] grows more rapidly than [latex]x^2[\/latex]. As [latex]x \\to \u2212\\infty, \\, x^2 \\to \\infty[\/latex], whereas [latex]2^x \\to 0[\/latex].<\/figcaption><\/figure>\n<\/section>\n<section class=\"textbox recall\">\n<table style=\"height: 240px;\">\n<thead>\n<tr style=\"height: 15px;\">\n<th style=\"text-align: center; height: 15px; width: 554.097px;\" colspan=\"2\">Arrow Notation<\/th>\n<\/tr>\n<tr style=\"height: 15px;\">\n<th style=\"text-align: center; height: 15px; width: 161.875px;\">Symbol<\/th>\n<th style=\"text-align: center; height: 15px; width: 380.764px;\">Meaning<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 161.875px;\">[latex]x\\to \\infty[\/latex]<\/td>\n<td style=\"height: 30px; width: 380.764px;\">[latex]x[\/latex] approaches infinity ([latex]x[\/latex]\u00a0increases without bound)<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 161.875px;\">[latex]x\\to -\\infty[\/latex]<\/td>\n<td style=\"height: 30px; width: 380.764px;\">[latex]x[\/latex] approaches negative infinity ([latex]x[\/latex]\u00a0decreases without bound)<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 161.875px;\">[latex]f\\left(x\\right)\\to \\infty[\/latex]<\/td>\n<td style=\"height: 30px; width: 380.764px;\">the output approaches infinity (the output increases without bound)<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 161.875px;\">[latex]f\\left(x\\right)\\to -\\infty[\/latex]<\/td>\n<td style=\"height: 30px; width: 380.764px;\">the output approaches negative infinity (the output decreases without bound)<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px; width: 161.875px;\">[latex]f\\left(x\\right)\\to a[\/latex]<\/td>\n<td style=\"height: 30px; width: 380.764px;\">the output approaches [latex]a[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n","protected":false},"author":6,"menu_order":17,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"1.5 Exponential and Logarithmic Functions\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":759,"module-header":"learn_it","content_attributions":[{"type":"cc","description":"Calculus Volume 1","author":"Gilbert Strang, Edwin (Jed) Herman","organization":"OpenStax","url":"https:\/\/openstax.org\/details\/books\/calculus-volume-1","project":"","license":"cc-by-nc-sa","license_terms":"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction"},{"type":"original","description":"1.5 Exponential and Logarithmic Functions","author":"Ryan Melton","organization":"","url":"","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/165"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":33,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/165\/revisions"}],"predecessor-version":[{"id":2395,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/165\/revisions\/2395"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/759"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/165\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=165"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=165"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=165"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=165"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}