{"id":3769,"date":"2021-05-12T20:24:06","date_gmt":"2021-05-12T20:24:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/1e\/"},"modified":"2021-05-12T20:24:06","modified_gmt":"2021-05-12T20:24:06","slug":"1e","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/1e\/","title":{"raw":"Skills Review for Exponential and Logarithmic Functions","rendered":"Skills Review for Exponential and Logarithmic Functions"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Model exponential growth and decay<\/li>\n<\/ul>\n<\/div>\nOne of the main topics covered in the Exponential and Logarithmic Functions section is graphing exponential functions. The ability to identify whether an exponential function represents exponential growth or decay, which is reviewed here, is an important aspect of determining the shape of its graph.\n<h2>Identify Exponential Growth and Decay<\/h2>\nIn real-world applications, we need to model the behavior of a function. In mathematical modeling, we choose a familiar general function with properties that suggest that it will model the real-world phenomenon we wish to analyze. In the case of rapid growth (or decay), we may choose to model the given scenario using the following function:\n<p style=\"text-align: center;\">[latex]y={A}_{0}{b}^{x}[\/latex]<\/p>\nwhere [latex]{A}_{0}[\/latex] is equal to the value at [latex]x=0[\/latex],&nbsp;[latex]b[\/latex] is the base, and [latex]x[\/latex]&nbsp;is the exponent. Note that the variable is in the exponent which makes the function exponential.\n\nWhen [latex]b&gt;0[\/latex], the exponential function represents&nbsp;<strong>exponential growt<\/strong><strong>h<\/strong>. Common applications of exponential growth include&nbsp;<strong>doubling time<\/strong>, the time it takes for a quantity to double. Such phenomena as wildlife populations, financial investments, biological samples, and natural resources may exhibit growth based on a doubling time.\n\nWhen [latex]b&lt;0[\/latex], the exponential function represents&nbsp;<strong>exponential decay<\/strong>. One common application of exponential decay includes&nbsp;calculating <strong>half-life<\/strong>,&nbsp;or the time it takes for a substance to exponentially decay to half of its original quantity. We use half-life in applications involving radioactive isotopes.\n\nIn our choice of a function to serve as a mathematical model, we often use data points gathered by careful observation and measurement to construct points on a graph and hope we can recognize the shape of the graph. Exponential growth and decay graphs have a distinctive shape, as we can see in the graphs below. It is important to remember that, although parts of each of the two graphs seem to lie on the <em>x<\/em>-axis, they are really a tiny distance above the <em>x<\/em>-axis.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03181337\/CNX_Precalc_Figure_04_07_0022.jpg\" alt=\"Graph of y=2e^(3x) with the labeled points (-1\/3, 2\/e), (0, 2), and (1\/3, 2e) and with the asymptote at y=0.\" width=\"487\" height=\"326\"> A graph showing exponential growth. The equation is [latex]y=2{e}^{3x}[\/latex].[\/caption][caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03181339\/CNX_Precalc_Figure_04_07_0032.jpg\" alt=\"Graph of y=3e^(-2x) with the labeled points (-1\/2, 3e), (0, 3), and (1\/2, 3\/e) and with the asymptote at y=0.\" width=\"487\" height=\"438\"> A graph showing exponential decay. The equation is [latex]y=3{e}^{-2x}[\/latex].[\/caption]\n<div class=\"textbox\">\n<h3>A General Note: Characteristics of the Exponential Function [latex]y=A_{0}b^{x}[\/latex]<\/h3>\nAn exponential function of the form [latex]y={A}_{0}{b}^{x}[\/latex] has the following characteristics:\n<ul>\n \t<li>one-to-one function<\/li>\n \t<li>horizontal asymptote: <em>y&nbsp;<\/em>= 0<\/li>\n \t<li>domain: [latex]\\left(-\\infty , \\infty \\right)[\/latex]<\/li>\n \t<li>range: [latex]\\left(0,\\infty \\right)[\/latex]<\/li>\n \t<li>x intercept: none<\/li>\n \t<li>y-intercept: [latex]\\left(0,{A}_{0}\\right)[\/latex]<\/li>\n \t<li>increasing if b<em>&nbsp;<\/em>&gt; 0<\/li>\n \t<li>decreasing if b<em>&nbsp;<\/em>&lt; 0<\/li>\n<\/ul>\nAn exponential function models exponential growth when [latex]b &gt; 0[\/latex] and exponential decay when [latex]b &lt; 0[\/latex].\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question hide_question_numbers=1]218951[\/ohm_question]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question hide_question_numbers=1]218952[\/ohm_question]\n\n<\/div>\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Model exponential growth and decay<\/li>\n<\/ul>\n<\/div>\n<p>One of the main topics covered in the Exponential and Logarithmic Functions section is graphing exponential functions. The ability to identify whether an exponential function represents exponential growth or decay, which is reviewed here, is an important aspect of determining the shape of its graph.<\/p>\n<h2>Identify Exponential Growth and Decay<\/h2>\n<p>In real-world applications, we need to model the behavior of a function. In mathematical modeling, we choose a familiar general function with properties that suggest that it will model the real-world phenomenon we wish to analyze. In the case of rapid growth (or decay), we may choose to model the given scenario using the following function:<\/p>\n<p style=\"text-align: center;\">[latex]y={A}_{0}{b}^{x}[\/latex]<\/p>\n<p>where [latex]{A}_{0}[\/latex] is equal to the value at [latex]x=0[\/latex],&nbsp;[latex]b[\/latex] is the base, and [latex]x[\/latex]&nbsp;is the exponent. Note that the variable is in the exponent which makes the function exponential.<\/p>\n<p>When [latex]b>0[\/latex], the exponential function represents&nbsp;<strong>exponential growt<\/strong><strong>h<\/strong>. Common applications of exponential growth include&nbsp;<strong>doubling time<\/strong>, the time it takes for a quantity to double. Such phenomena as wildlife populations, financial investments, biological samples, and natural resources may exhibit growth based on a doubling time.<\/p>\n<p>When [latex]b<0[\/latex], the exponential function represents&nbsp;<strong>exponential decay<\/strong>. One common application of exponential decay includes&nbsp;calculating <strong>half-life<\/strong>,&nbsp;or the time it takes for a substance to exponentially decay to half of its original quantity. We use half-life in applications involving radioactive isotopes.<\/p>\n<p>In our choice of a function to serve as a mathematical model, we often use data points gathered by careful observation and measurement to construct points on a graph and hope we can recognize the shape of the graph. Exponential growth and decay graphs have a distinctive shape, as we can see in the graphs below. It is important to remember that, although parts of each of the two graphs seem to lie on the <em>x<\/em>-axis, they are really a tiny distance above the <em>x<\/em>-axis.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03181337\/CNX_Precalc_Figure_04_07_0022.jpg\" alt=\"Graph of y=2e^(3x) with the labeled points (-1\/3, 2\/e), (0, 2), and (1\/3, 2e) and with the asymptote at y=0.\" width=\"487\" height=\"326\" \/><\/p>\n<p class=\"wp-caption-text\">A graph showing exponential growth. The equation is [latex]y=2{e}^{3x}[\/latex].<\/p>\n<\/div>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03181339\/CNX_Precalc_Figure_04_07_0032.jpg\" alt=\"Graph of y=3e^(-2x) with the labeled points (-1\/2, 3e), (0, 3), and (1\/2, 3\/e) and with the asymptote at y=0.\" width=\"487\" height=\"438\" \/><\/p>\n<p class=\"wp-caption-text\">A graph showing exponential decay. The equation is [latex]y=3{e}^{-2x}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Characteristics of the Exponential Function [latex]y=A_{0}b^{x}[\/latex]<\/h3>\n<p>An exponential function of the form [latex]y={A}_{0}{b}^{x}[\/latex] has the following characteristics:<\/p>\n<ul>\n<li>one-to-one function<\/li>\n<li>horizontal asymptote: <em>y&nbsp;<\/em>= 0<\/li>\n<li>domain: [latex]\\left(-\\infty , \\infty \\right)[\/latex]<\/li>\n<li>range: [latex]\\left(0,\\infty \\right)[\/latex]<\/li>\n<li>x intercept: none<\/li>\n<li>y-intercept: [latex]\\left(0,{A}_{0}\\right)[\/latex]<\/li>\n<li>increasing if b<em>&nbsp;<\/em>&gt; 0<\/li>\n<li>decreasing if b<em>&nbsp;<\/em>&lt; 0<\/li>\n<\/ul>\n<p>An exponential function models exponential growth when [latex]b > 0[\/latex] and exponential decay when [latex]b < 0[\/latex].\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm218951\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=218951&theme=oea&iframe_resize_id=ohm218951\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm218952\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=218952&theme=oea&iframe_resize_id=ohm218952\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-3769\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Modification and Revision. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"College Algebra Corequisite\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/precalculus\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Modification and Revision\",\"author\":\"\",\"organization\":\"Lumen 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