{"id":5337,"date":"2021-10-13T18:24:23","date_gmt":"2021-10-13T18:24:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/chapter\/introduction-exponential-functions\/"},"modified":"2024-07-26T15:54:13","modified_gmt":"2024-07-26T15:54:13","slug":"introduction-exponential-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/chapter\/introduction-exponential-functions\/","title":{"raw":"Exponential Functions","rendered":"Exponential Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify the base of an exponential function and restrictions for its value.<\/li>\r\n \t<li>Evaluate exponential functions.<\/li>\r\n \t<li>Evaluate exponential functions with base e.<\/li>\r\n \t<li>Investigate continuous growth and decay.<\/li>\r\n \t<li>Determine whether an exponential function and its associated graph represents growth or decay.<\/li>\r\n \t<li>Sketch a graph of an exponential function.<\/li>\r\n \t<li>Introduction to the logistic growth model<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165137540105\">Focus in on a square centimeter of your skin. Look closer. Closer still. If you could look closely enough, you would see hundreds of thousands of microscopic organisms. They are bacteria, and they are not only on your skin, but in your mouth, nose, and even your intestines. In fact, the bacterial cells in your body at any given moment outnumber your own cells. But that is no reason to feel bad about yourself. While some bacteria can cause illness, many are healthy and even essential to the body.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02225437\/CNX_Precalc_Figure_04_00_001.jpg\" alt=\"Escherichia coli (e Coli) bacteria\" width=\"975\" height=\"704\" \/> An electron micrograph of E.Coli bacteria. (credit: \u201cMattosaurus,\u201d Wikimedia Commons)[\/caption]\r\n<p id=\"fs-id1165135456742\">Bacteria commonly reproduce through a process called binary fission during which one bacterial cell splits into two. When conditions are right, bacteria can reproduce very quickly. Unlike humans and other complex organisms, the time required to form a new generation of bacteria is often a matter of minutes or hours as opposed to days or years.[footnote]Todar, PhD, Kenneth. Todar's Online Textbook of Bacteriology. <a href=\"http:\/\/textbookofbacteriology.net\/growth_3.html\" target=\"_blank\" rel=\"noopener\">http:\/\/textbookofbacteriology.net\/growth_3.html<\/a>.[\/footnote]<\/p>\r\n<p id=\"fs-id1165135547348\">For simplicity\u2019s sake, suppose we begin with a culture of one bacterial cell that can divide every hour. The table below\u00a0shows the number of bacterial cells at the end of each subsequent hour. We see that the single bacterial cell leads to over one thousand bacterial cells in just ten hours! If we were to extrapolate the table to twenty-four hours, we would have over 16 million!<\/p>\r\n\r\n<table id=\"Table_04_00_01\" summary=\"\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Hour<\/strong><\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<td>8<\/td>\r\n<td>9<\/td>\r\n<td>10<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Bacteria<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>8<\/td>\r\n<td>16<\/td>\r\n<td>32<\/td>\r\n<td>64<\/td>\r\n<td>128<\/td>\r\n<td>256<\/td>\r\n<td>512<\/td>\r\n<td>1024<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn this unit, we will explore exponential functions which can be used for, among other things, modeling growth patterns such as those found in bacteria. We will also investigate logarithmic functions which are closely related to exponential functions. Both types of functions have numerous real-world applications when it comes to modeling and interpreting data.\r\n<h2>Evaluating Exponential Functions<\/h2>\r\nThe base of an exponential function must be a positive real number other than 1. Why do we limit the base <em>b<\/em>\u00a0to positive values? This is done to ensure that the outputs will be real numbers. Observe what happens if the base is not positive:\r\n<ul>\r\n \t<li>Consider a base of \u20139 and exponent of [latex]\\frac{1}{2}[\/latex]. Then [latex]f\\left(x\\right)=f\\left(\\frac{1}{2}\\right)={\\left(-9\\right)}^{\\frac{1}{2}}=\\sqrt{-9}[\/latex], which is not a real number.<\/li>\r\n<\/ul>\r\nWhy do we limit the base to positive values other than 1? This is because a base of 1\u00a0results in the constant function. Observe what happens if the base is\u00a01:\r\n<ul>\r\n \t<li>Consider a base of 1.\u00a0Then [latex]f\\left(x\\right)={1}^{x}=1[\/latex] for any value of <em>x<\/em>.<\/li>\r\n<\/ul>\r\nTo evaluate an exponential function with the form [latex]f\\left(x\\right)={b}^{x}[\/latex], we simply substitute <em>x<\/em>\u00a0with the given value, and calculate the resulting power. For example:\r\n\r\nLet [latex]f\\left(x\\right)={2}^{x}[\/latex]. What is [latex]f\\left(3\\right)[\/latex]?\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{llllllll}f\\left(x\\right)\\hfill &amp; ={2}^{x}\\hfill &amp; \\hfill \\\\ f\\left(3\\right)\\hfill &amp; ={2}^{3}\\text{}\\hfill &amp; \\text{Substitute }x=3. \\hfill \\\\ \\hfill &amp; =8\\text{}\\hfill &amp; \\text{Evaluate the power}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\r\nTo evaluate an exponential function, it is important to follow the order of operations. For example:\r\n\r\nLet [latex]f\\left(x\\right)=30{\\left(2\\right)}^{x}[\/latex]. What is [latex]f\\left(3\\right)[\/latex]?\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)\\hfill &amp; =30{\\left(2\\right)}^{x}\\hfill &amp; \\hfill \\\\ f\\left(3\\right)\\hfill &amp; =30{\\left(2\\right)}^{3}\\hfill &amp; \\text{Substitute }x=3.\\hfill \\\\ \\hfill &amp; =30\\left(8\\right)\\text{ }\\hfill &amp; \\text{Simplify the power first}\\text{.}\\hfill \\\\ \\hfill &amp; =240\\hfill &amp; \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\r\nNote that if the order of operations were not followed, the result would be incorrect:\r\n<p style=\"text-align: center;\">[latex]f\\left(3\\right)=30{\\left(2\\right)}^{3}\\ne {60}^{3}=216,000[\/latex]<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating Exponential Functions<\/h3>\r\nLet [latex]f\\left(x\\right)=5{\\left(3\\right)}^{x+1}[\/latex]. Evaluate [latex]f\\left(2\\right)[\/latex] without using a calculator.\r\n\r\n[reveal-answer q=\"454575\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"454575\"]\r\n\r\nFollow the order of operations. Be sure to pay attention to the parentheses.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)\\hfill &amp; =5{\\left(3\\right)}^{x+1}\\hfill &amp; \\hfill \\\\ f\\left(2\\right)\\hfill &amp; =5{\\left(3\\right)}^{2+1}\\hfill &amp; \\text{Substitute }x=2.\\hfill \\\\ \\hfill &amp; =5{\\left(3\\right)}^{3}\\hfill &amp; \\text{Add the exponents}.\\hfill \\\\ \\hfill &amp; =5\\left(27\\right)\\hfill &amp; \\text{Simplify the power}\\text{.}\\hfill \\\\ \\hfill &amp; =135\\hfill &amp; \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nLet [latex]f\\left(x\\right)=8{\\left(1.2\\right)}^{x - 5}[\/latex]. Evaluate [latex]f\\left(3\\right)[\/latex] using a calculator. Round to four decimal places.\r\n\r\n[reveal-answer q=\"860098\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"860098\"]\r\n\r\n5.5556[\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=73212&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\">\r\n<\/iframe>\r\n\r\n<\/div>\r\nBecause the output of exponential functions increases very rapidly, the term \"exponential growth\" is often used in everyday language to describe anything that grows or increases rapidly. However, exponential growth can be defined more precisely in a mathematical sense. If the growth rate is proportional to the amount present, the function models exponential growth.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Exponential Growth<\/h3>\r\nA function that models <strong>exponential growth<\/strong> grows by a rate, <em>r<\/em>,\u00a0 proportional to the amount present. For any real number <em>x<\/em>\u00a0and any positive real numbers <em>a\u00a0<\/em>and <em>b<\/em>\u00a0such that [latex]b\\ne 1[\/latex], an exponential growth function has the form\r\n\r\n[latex]\\text{ }f\\left(x\\right)=a{b}^{x}[\/latex]\r\n\r\nwhere\r\n<ul>\r\n \t<li><em>a<\/em>\u00a0is the initial or starting value of the function.<\/li>\r\n \t<li><em>b<\/em>\u00a0is the growth factor or growth multiplier per unit <em>x<\/em>.\u00a0 The growth factor is based on the growth rate, r.\u00a0 The value of the growth factor can be found by using <em>b = 1+r<\/em>.\u00a0 Note, if you have exponential decay and are given the decay rate, <em>r<\/em>, then <em>b = 1 - r.<\/em><\/li>\r\n<\/ul>\r\n<\/div>\r\nIn more general terms, an <em>exponential function <\/em>consists of a\u00a0constant base raised to a variable exponent. To differentiate between linear and exponential functions, let\u2019s consider two companies, A and B. Company A has 100 stores and expands by opening 50 new stores a year, so its growth can be represented by the function [latex]A\\left(x\\right)=100+50x[\/latex]. Company B has 100 stores and expands by increasing the number of stores by 50% each year, so its growth can be represented by the function [latex]B\\left(x\\right)=100{\\left(1+0.5\\right)}^{x}[\/latex].\r\n\r\nA few years of growth for these companies are illustrated below.\r\n<table summary=\"Six rows and three columns. The first column is labeled,\">\r\n<thead>\r\n<tr>\r\n<th>Year,\u00a0<em>x<\/em><\/th>\r\n<th>Stores, Company A<\/th>\r\n<th>Stores, Company B<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>0<\/td>\r\n<td>100 + 50(0) = 100<\/td>\r\n<td>100(1 + 0.5)<sup>0<\/sup> = 100<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td>100 + 50(1) = 150<\/td>\r\n<td>100(1 + 0.5)<sup>1<\/sup> = 150<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>100 + 50(2) = 200<\/td>\r\n<td>100(1 + 0.5)<sup>2<\/sup> = 225<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>100 + 50(3) = 250<\/td>\r\n<td>100(1 + 0.5)<sup>3<\/sup> =\u00a0337.5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>x<\/em><\/td>\r\n<td><em>A<\/em>(<em>x<\/em>) = 100 + 50x<\/td>\r\n<td><em>B<\/em>(<em>x<\/em>) = 100(1 + 0.5)<sup><em>x<\/em><\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe graphs comparing the number of stores for each company over a five-year period are shown below. We can see that, with exponential growth, the number of stores increases much more rapidly than with linear growth.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02225441\/CNX_Precalc_Figure_04_01_0012.jpg\" alt=\"Graph of Companies A and B\u2019s functions, which values are found in the previous table.\" width=\"487\" height=\"843\" \/> The graph shows the numbers of stores Companies A and B opened over a five-year period.[\/caption]\r\n\r\nNotice that the domain for both functions is [latex]\\left[0,\\infty \\right)[\/latex], and the range for both functions is [latex]\\left[100,\\infty \\right)[\/latex]. After year 1, Company B always has more stores than Company A.\r\n\r\n&nbsp;\r\n<h3>Evaluating Exponential Functions with Base e<\/h3>\r\nAs <em>n<\/em>\u00a0gets larger and larger, the expression [latex]{\\left(1+\\frac{1}{n}\\right)}^{n}[\/latex] approaches a number used so frequently in mathematics that it has its own name: the letter [latex]e[\/latex]. This value is an irrational number, which means that its decimal expansion goes on forever without repeating. Its approximation to six decimal places is shown below.\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Number [latex]e[\/latex]<\/h3>\r\nThe letter <em>e<\/em> represents the irrational number\r\n<p style=\"text-align: center;\">[latex]{\\left(1+\\frac{1}{n}\\right)}^{n},\\text{as }n\\text{ increases without bound}[\/latex]<\/p>\r\nThe letter <em>e <\/em>is used as a base for many real-world exponential models. To work with base <em>e<\/em>, we use the approximation, [latex]e\\approx 2.718282[\/latex]. The constant was named by the Swiss mathematician Leonhard Euler (1707\u20131783) who first investigated and discovered many of its properties.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using a Calculator to Find Powers of [latex]e[\/latex]<\/h3>\r\nCalculate [latex]{e}^{3.14}[\/latex]. Round to five decimal places.\r\n\r\n[reveal-answer q=\"201253\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"201253\"]\r\n\r\nOn a calculator, press the button labeled [latex]\\left[{e}^{x}\\right][\/latex]. The window shows [<em>e<\/em>^(]. Type 3.14 and then close parenthesis, (]). Press [ENTER]. Rounding to 5 decimal places, [latex]{e}^{3.14}\\approx 23.10387[\/latex]. Caution: Many scientific calculators have an \"Exp\" button, which is used to enter numbers in scientific notation. It is not used to find powers of <em>e<\/em>.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse a calculator to find [latex]{e}^{-0.5}[\/latex]. Round to five decimal places.\r\n\r\n[reveal-answer q=\"168744\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"168744\"]\r\n\r\n[latex]{e}^{-0.5}\\approx 0.60653[\/latex][\/hidden-answer]\r\n<iframe id=\"mom11\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1495&amp;theme=oea&amp;iframe_resize_id=mom11\" width=\"100%\" height=\"350\">\r\n<\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<h3>Investigating Continuous Growth<\/h3>\r\nSo far we have worked with rational bases for exponential functions. For most real-world phenomena, however, <em>e <\/em>is used as the base for exponential functions. Exponential models that use <em>e<\/em>\u00a0as the base are called <em>continuous growth or decay models<\/em>. We see these models in finance, computer science, and most of the sciences such as physics, toxicology, and fluid dynamics.\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Continuous Growth\/Decay Formula<\/h3>\r\nFor all real numbers <em>t<\/em>, and all positive numbers <em>a<\/em>\u00a0and <em>r<\/em>, continuous growth or decay is represented by the formula\r\n<p style=\"text-align: center;\">[latex]A\\left(t\\right)=a{e}^{rt}[\/latex]<\/p>\r\nwhere\r\n<ul>\r\n \t<li><em>a<\/em>\u00a0is the initial value<\/li>\r\n \t<li><em>r<\/em>\u00a0is the continuous growth rate per unit of time<\/li>\r\n \t<li><em>t<\/em>\u00a0is the elapsed time<\/li>\r\n<\/ul>\r\nIf <em>r\u00a0<\/em>&gt; 0, then the formula represents continuous growth. If <em>r\u00a0<\/em>&lt; 0, then the formula represents continuous decay.\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nFor business applications, the continuous growth formula is called the continuous compounding formula and takes the form\r\n<p style=\"text-align: center;\">[latex]A\\left(t\\right)=P{e}^{rt}[\/latex]<\/p>\r\nwhere\r\n<ul>\r\n \t<li><em>P<\/em>\u00a0is the principal or the initial investment<\/li>\r\n \t<li><em>r<\/em>\u00a0is the growth or interest rate per unit of time<\/li>\r\n \t<li><em>t<\/em>\u00a0is the period or term of the investment<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given the initial value, rate of growth or decay, and time <em>t<\/em>, solve a continuous growth or decay function<\/h3>\r\n<ol>\r\n \t<li>Use the information in the problem to determine <em>a<\/em>, the initial value of the function.<\/li>\r\n \t<li>Use the information in the problem to determine the growth rate <em>r<\/em>.\r\n<ul>\r\n \t<li>If the problem refers to continuous growth, then <em>r\u00a0<\/em>&gt; 0.<\/li>\r\n \t<li>If the problem refers to continuous decay, then <em>r\u00a0<\/em>&lt; 0.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Use the information in the problem to determine the time <em>t<\/em>.<\/li>\r\n \t<li>Substitute the given information into the continuous growth formula and solve for <em>A<\/em>(<em>t<\/em>).<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Calculating Continuous Growth<\/h3>\r\nA person invested $1,000 in an account earning a nominal interest rate of 10% per year compounded continuously. How much was in the account at the end of one year?\r\n\r\n[reveal-answer q=\"864251\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"864251\"]\r\n\r\nSince the account is growing in value, this is a continuous compounding problem with growth rate <em>r\u00a0<\/em>= 0.10. The initial investment was $1,000, so <em>P\u00a0<\/em>= 1000. We use the continuous compounding formula to find the value after <em>t\u00a0<\/em>= 1 year:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}A\\left(t\\right)\\hfill &amp; =P{e}^{rt}\\hfill &amp; \\text{Use the continuous compounding formula}.\\hfill \\\\ \\hfill &amp; =1000{\\left(e\\right)}^{0.1} &amp; \\text{Substitute known values for }P, r,\\text{ and }t.\\hfill \\\\ \\hfill &amp; \\approx 1105.17\\hfill &amp; \\text{Use a calculator to approximate}.\\hfill \\end{array}[\/latex]<\/p>\r\nThe account is worth $1,105.17 after one year.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nA person invests $100,000 at a nominal 12% interest per year compounded continuously. What will be the value of the investment in 30 years?\r\n\r\n[reveal-answer q=\"59872\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"59872\"]\r\n\r\n$3,659,823.44[\/hidden-answer]\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=25526&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"300\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Calculating Continuous Decay<\/h3>\r\nRadon-222 decays at a continuous rate of 17.3% per day. How much will 100 mg of Radon-222 decay to in 3 days?\r\n\r\n[reveal-answer q=\"660828\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"660828\"]\r\n\r\nSince the substance is decaying, the rate, 17.3%, is negative. So, <em>r\u00a0<\/em>=\u00a0\u20130.173. The initial amount of radon-222 was 100 mg, so <em>a\u00a0<\/em>= 100. We use the continuous decay formula to find the value after <em>t\u00a0<\/em>= 3 days:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}A\\left(t\\right)\\hfill &amp; =a{e}^{rt}\\hfill &amp; \\text{Use the continuous growth formula}.\\hfill \\\\ \\hfill &amp; =100{e}^{-0.173\\left(3\\right)} &amp; \\text{Substitute known values for }a, r,\\text{ and }t.\\hfill \\\\ \\hfill &amp; \\approx 59.5115\\hfill &amp; \\text{Use a calculator to approximate}.\\hfill \\end{array}[\/latex]<\/p>\r\nSo 59.5115 mg of radon-222 will remain.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUsing the data in the previous example, how much radon-222 will remain after one year?\r\n\r\n[reveal-answer q=\"58534\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"58534\"]\r\n\r\n3.77E-26 (This is calculator notation for the number written as [latex]3.77\\times {10}^{-26}[\/latex] in scientific notation. While the output of an exponential function is never zero, this number is so close to zero that for all practical purposes we can accept zero as the answer.)[\/hidden-answer]\r\n\r\n<\/div>\r\n<section id=\"fs-id1165135264762\" class=\"key-equations\"><\/section>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify the base of an exponential function and restrictions for its value.<\/li>\n<li>Evaluate exponential functions.<\/li>\n<li>Evaluate exponential functions with base e.<\/li>\n<li>Investigate continuous growth and decay.<\/li>\n<li>Determine whether an exponential function and its associated graph represents growth or decay.<\/li>\n<li>Sketch a graph of an exponential function.<\/li>\n<li>Introduction to the logistic growth model<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137540105\">Focus in on a square centimeter of your skin. Look closer. Closer still. If you could look closely enough, you would see hundreds of thousands of microscopic organisms. They are bacteria, and they are not only on your skin, but in your mouth, nose, and even your intestines. In fact, the bacterial cells in your body at any given moment outnumber your own cells. But that is no reason to feel bad about yourself. While some bacteria can cause illness, many are healthy and even essential to the body.<\/p>\n<div style=\"width: 985px\" 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\/02225437\/CNX_Precalc_Figure_04_00_001.jpg\" alt=\"Escherichia coli (e Coli) bacteria\" width=\"975\" height=\"704\" \/><\/p>\n<p class=\"wp-caption-text\">An electron micrograph of E.Coli bacteria. (credit: \u201cMattosaurus,\u201d Wikimedia Commons)<\/p>\n<\/div>\n<p id=\"fs-id1165135456742\">Bacteria commonly reproduce through a process called binary fission during which one bacterial cell splits into two. When conditions are right, bacteria can reproduce very quickly. Unlike humans and other complex organisms, the time required to form a new generation of bacteria is often a matter of minutes or hours as opposed to days or years.<a class=\"footnote\" title=\"Todar, PhD, Kenneth. Todar's Online Textbook of Bacteriology. http:\/\/textbookofbacteriology.net\/growth_3.html.\" id=\"return-footnote-5337-1\" href=\"#footnote-5337-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<p id=\"fs-id1165135547348\">For simplicity\u2019s sake, suppose we begin with a culture of one bacterial cell that can divide every hour. The table below\u00a0shows the number of bacterial cells at the end of each subsequent hour. We see that the single bacterial cell leads to over one thousand bacterial cells in just ten hours! If we were to extrapolate the table to twenty-four hours, we would have over 16 million!<\/p>\n<table id=\"Table_04_00_01\" summary=\"\">\n<tbody>\n<tr>\n<td><strong>Hour<\/strong><\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>8<\/td>\n<td>9<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td><strong>Bacteria<\/strong><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>8<\/td>\n<td>16<\/td>\n<td>32<\/td>\n<td>64<\/td>\n<td>128<\/td>\n<td>256<\/td>\n<td>512<\/td>\n<td>1024<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In this unit, we will explore exponential functions which can be used for, among other things, modeling growth patterns such as those found in bacteria. We will also investigate logarithmic functions which are closely related to exponential functions. Both types of functions have numerous real-world applications when it comes to modeling and interpreting data.<\/p>\n<h2>Evaluating Exponential Functions<\/h2>\n<p>The base of an exponential function must be a positive real number other than 1. Why do we limit the base <em>b<\/em>\u00a0to positive values? This is done to ensure that the outputs will be real numbers. Observe what happens if the base is not positive:<\/p>\n<ul>\n<li>Consider a base of \u20139 and exponent of [latex]\\frac{1}{2}[\/latex]. Then [latex]f\\left(x\\right)=f\\left(\\frac{1}{2}\\right)={\\left(-9\\right)}^{\\frac{1}{2}}=\\sqrt{-9}[\/latex], which is not a real number.<\/li>\n<\/ul>\n<p>Why do we limit the base to positive values other than 1? This is because a base of 1\u00a0results in the constant function. Observe what happens if the base is\u00a01:<\/p>\n<ul>\n<li>Consider a base of 1.\u00a0Then [latex]f\\left(x\\right)={1}^{x}=1[\/latex] for any value of <em>x<\/em>.<\/li>\n<\/ul>\n<p>To evaluate an exponential function with the form [latex]f\\left(x\\right)={b}^{x}[\/latex], we simply substitute <em>x<\/em>\u00a0with the given value, and calculate the resulting power. For example:<\/p>\n<p>Let [latex]f\\left(x\\right)={2}^{x}[\/latex]. What is [latex]f\\left(3\\right)[\/latex]?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{llllllll}f\\left(x\\right)\\hfill & ={2}^{x}\\hfill & \\hfill \\\\ f\\left(3\\right)\\hfill & ={2}^{3}\\text{}\\hfill & \\text{Substitute }x=3. \\hfill \\\\ \\hfill & =8\\text{}\\hfill & \\text{Evaluate the power}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\n<p>To evaluate an exponential function, it is important to follow the order of operations. For example:<\/p>\n<p>Let [latex]f\\left(x\\right)=30{\\left(2\\right)}^{x}[\/latex]. What is [latex]f\\left(3\\right)[\/latex]?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)\\hfill & =30{\\left(2\\right)}^{x}\\hfill & \\hfill \\\\ f\\left(3\\right)\\hfill & =30{\\left(2\\right)}^{3}\\hfill & \\text{Substitute }x=3.\\hfill \\\\ \\hfill & =30\\left(8\\right)\\text{ }\\hfill & \\text{Simplify the power first}\\text{.}\\hfill \\\\ \\hfill & =240\\hfill & \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\n<p>Note that if the order of operations were not followed, the result would be incorrect:<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(3\\right)=30{\\left(2\\right)}^{3}\\ne {60}^{3}=216,000[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating Exponential Functions<\/h3>\n<p>Let [latex]f\\left(x\\right)=5{\\left(3\\right)}^{x+1}[\/latex]. Evaluate [latex]f\\left(2\\right)[\/latex] without using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q454575\">Show Solution<\/span><\/p>\n<div id=\"q454575\" class=\"hidden-answer\" style=\"display: none\">\n<p>Follow the order of operations. Be sure to pay attention to the parentheses.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)\\hfill & =5{\\left(3\\right)}^{x+1}\\hfill & \\hfill \\\\ f\\left(2\\right)\\hfill & =5{\\left(3\\right)}^{2+1}\\hfill & \\text{Substitute }x=2.\\hfill \\\\ \\hfill & =5{\\left(3\\right)}^{3}\\hfill & \\text{Add the exponents}.\\hfill \\\\ \\hfill & =5\\left(27\\right)\\hfill & \\text{Simplify the power}\\text{.}\\hfill \\\\ \\hfill & =135\\hfill & \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Let [latex]f\\left(x\\right)=8{\\left(1.2\\right)}^{x - 5}[\/latex]. Evaluate [latex]f\\left(3\\right)[\/latex] using a calculator. Round to four decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q860098\">Show Solution<\/span><\/p>\n<div id=\"q860098\" class=\"hidden-answer\" style=\"display: none\">\n<p>5.5556<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=73212&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><br \/>\n<\/iframe><\/p>\n<\/div>\n<p>Because the output of exponential functions increases very rapidly, the term &#8220;exponential growth&#8221; is often used in everyday language to describe anything that grows or increases rapidly. However, exponential growth can be defined more precisely in a mathematical sense. If the growth rate is proportional to the amount present, the function models exponential growth.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Exponential Growth<\/h3>\n<p>A function that models <strong>exponential growth<\/strong> grows by a rate, <em>r<\/em>,\u00a0 proportional to the amount present. For any real number <em>x<\/em>\u00a0and any positive real numbers <em>a\u00a0<\/em>and <em>b<\/em>\u00a0such that [latex]b\\ne 1[\/latex], an exponential growth function has the form<\/p>\n<p>[latex]\\text{ }f\\left(x\\right)=a{b}^{x}[\/latex]<\/p>\n<p>where<\/p>\n<ul>\n<li><em>a<\/em>\u00a0is the initial or starting value of the function.<\/li>\n<li><em>b<\/em>\u00a0is the growth factor or growth multiplier per unit <em>x<\/em>.\u00a0 The growth factor is based on the growth rate, r.\u00a0 The value of the growth factor can be found by using <em>b = 1+r<\/em>.\u00a0 Note, if you have exponential decay and are given the decay rate, <em>r<\/em>, then <em>b = 1 &#8211; r.<\/em><\/li>\n<\/ul>\n<\/div>\n<p>In more general terms, an <em>exponential function <\/em>consists of a\u00a0constant base raised to a variable exponent. To differentiate between linear and exponential functions, let\u2019s consider two companies, A and B. Company A has 100 stores and expands by opening 50 new stores a year, so its growth can be represented by the function [latex]A\\left(x\\right)=100+50x[\/latex]. Company B has 100 stores and expands by increasing the number of stores by 50% each year, so its growth can be represented by the function [latex]B\\left(x\\right)=100{\\left(1+0.5\\right)}^{x}[\/latex].<\/p>\n<p>A few years of growth for these companies are illustrated below.<\/p>\n<table summary=\"Six rows and three columns. The first column is labeled,\">\n<thead>\n<tr>\n<th>Year,\u00a0<em>x<\/em><\/th>\n<th>Stores, Company A<\/th>\n<th>Stores, Company B<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>100 + 50(0) = 100<\/td>\n<td>100(1 + 0.5)<sup>0<\/sup> = 100<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>100 + 50(1) = 150<\/td>\n<td>100(1 + 0.5)<sup>1<\/sup> = 150<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>100 + 50(2) = 200<\/td>\n<td>100(1 + 0.5)<sup>2<\/sup> = 225<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>100 + 50(3) = 250<\/td>\n<td>100(1 + 0.5)<sup>3<\/sup> =\u00a0337.5<\/td>\n<\/tr>\n<tr>\n<td><em>x<\/em><\/td>\n<td><em>A<\/em>(<em>x<\/em>) = 100 + 50x<\/td>\n<td><em>B<\/em>(<em>x<\/em>) = 100(1 + 0.5)<sup><em>x<\/em><\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The graphs comparing the number of stores for each company over a five-year period are shown below. We can see that, with exponential growth, the number of stores increases much more rapidly than with linear growth.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02225441\/CNX_Precalc_Figure_04_01_0012.jpg\" alt=\"Graph of Companies A and B\u2019s functions, which values are found in the previous table.\" width=\"487\" height=\"843\" \/><\/p>\n<p class=\"wp-caption-text\">The graph shows the numbers of stores Companies A and B opened over a five-year period.<\/p>\n<\/div>\n<p>Notice that the domain for both functions is [latex]\\left[0,\\infty \\right)[\/latex], and the range for both functions is [latex]\\left[100,\\infty \\right)[\/latex]. After year 1, Company B always has more stores than Company A.<\/p>\n<p>&nbsp;<\/p>\n<h3>Evaluating Exponential Functions with Base e<\/h3>\n<p>As <em>n<\/em>\u00a0gets larger and larger, the expression [latex]{\\left(1+\\frac{1}{n}\\right)}^{n}[\/latex] approaches a number used so frequently in mathematics that it has its own name: the letter [latex]e[\/latex]. This value is an irrational number, which means that its decimal expansion goes on forever without repeating. Its approximation to six decimal places is shown below.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Number [latex]e[\/latex]<\/h3>\n<p>The letter <em>e<\/em> represents the irrational number<\/p>\n<p style=\"text-align: center;\">[latex]{\\left(1+\\frac{1}{n}\\right)}^{n},\\text{as }n\\text{ increases without bound}[\/latex]<\/p>\n<p>The letter <em>e <\/em>is used as a base for many real-world exponential models. To work with base <em>e<\/em>, we use the approximation, [latex]e\\approx 2.718282[\/latex]. The constant was named by the Swiss mathematician Leonhard Euler (1707\u20131783) who first investigated and discovered many of its properties.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using a Calculator to Find Powers of [latex]e[\/latex]<\/h3>\n<p>Calculate [latex]{e}^{3.14}[\/latex]. Round to five decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q201253\">Show Solution<\/span><\/p>\n<div id=\"q201253\" class=\"hidden-answer\" style=\"display: none\">\n<p>On a calculator, press the button labeled [latex]\\left[{e}^{x}\\right][\/latex]. The window shows [<em>e<\/em>^(]. Type 3.14 and then close parenthesis, (]). Press [ENTER]. Rounding to 5 decimal places, [latex]{e}^{3.14}\\approx 23.10387[\/latex]. Caution: Many scientific calculators have an &#8220;Exp&#8221; button, which is used to enter numbers in scientific notation. It is not used to find powers of <em>e<\/em>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use a calculator to find [latex]{e}^{-0.5}[\/latex]. Round to five decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q168744\">Show Solution<\/span><\/p>\n<div id=\"q168744\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{e}^{-0.5}\\approx 0.60653[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom11\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1495&amp;theme=oea&amp;iframe_resize_id=mom11\" width=\"100%\" height=\"350\"><br \/>\n<\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>Investigating Continuous Growth<\/h3>\n<p>So far we have worked with rational bases for exponential functions. For most real-world phenomena, however, <em>e <\/em>is used as the base for exponential functions. Exponential models that use <em>e<\/em>\u00a0as the base are called <em>continuous growth or decay models<\/em>. We see these models in finance, computer science, and most of the sciences such as physics, toxicology, and fluid dynamics.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Continuous Growth\/Decay Formula<\/h3>\n<p>For all real numbers <em>t<\/em>, and all positive numbers <em>a<\/em>\u00a0and <em>r<\/em>, continuous growth or decay is represented by the formula<\/p>\n<p style=\"text-align: center;\">[latex]A\\left(t\\right)=a{e}^{rt}[\/latex]<\/p>\n<p>where<\/p>\n<ul>\n<li><em>a<\/em>\u00a0is the initial value<\/li>\n<li><em>r<\/em>\u00a0is the continuous growth rate per unit of time<\/li>\n<li><em>t<\/em>\u00a0is the elapsed time<\/li>\n<\/ul>\n<p>If <em>r\u00a0<\/em>&gt; 0, then the formula represents continuous growth. If <em>r\u00a0<\/em>&lt; 0, then the formula represents continuous decay.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>For business applications, the continuous growth formula is called the continuous compounding formula and takes the form<\/p>\n<p style=\"text-align: center;\">[latex]A\\left(t\\right)=P{e}^{rt}[\/latex]<\/p>\n<p>where<\/p>\n<ul>\n<li><em>P<\/em>\u00a0is the principal or the initial investment<\/li>\n<li><em>r<\/em>\u00a0is the growth or interest rate per unit of time<\/li>\n<li><em>t<\/em>\u00a0is the period or term of the investment<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the initial value, rate of growth or decay, and time <em>t<\/em>, solve a continuous growth or decay function<\/h3>\n<ol>\n<li>Use the information in the problem to determine <em>a<\/em>, the initial value of the function.<\/li>\n<li>Use the information in the problem to determine the growth rate <em>r<\/em>.\n<ul>\n<li>If the problem refers to continuous growth, then <em>r\u00a0<\/em>&gt; 0.<\/li>\n<li>If the problem refers to continuous decay, then <em>r\u00a0<\/em>&lt; 0.<\/li>\n<\/ul>\n<\/li>\n<li>Use the information in the problem to determine the time <em>t<\/em>.<\/li>\n<li>Substitute the given information into the continuous growth formula and solve for <em>A<\/em>(<em>t<\/em>).<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Calculating Continuous Growth<\/h3>\n<p>A person invested $1,000 in an account earning a nominal interest rate of 10% per year compounded continuously. How much was in the account at the end of one year?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q864251\">Show Solution<\/span><\/p>\n<div id=\"q864251\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since the account is growing in value, this is a continuous compounding problem with growth rate <em>r\u00a0<\/em>= 0.10. The initial investment was $1,000, so <em>P\u00a0<\/em>= 1000. We use the continuous compounding formula to find the value after <em>t\u00a0<\/em>= 1 year:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}A\\left(t\\right)\\hfill & =P{e}^{rt}\\hfill & \\text{Use the continuous compounding formula}.\\hfill \\\\ \\hfill & =1000{\\left(e\\right)}^{0.1} & \\text{Substitute known values for }P, r,\\text{ and }t.\\hfill \\\\ \\hfill & \\approx 1105.17\\hfill & \\text{Use a calculator to approximate}.\\hfill \\end{array}[\/latex]<\/p>\n<p>The account is worth $1,105.17 after one year.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A person invests $100,000 at a nominal 12% interest per year compounded continuously. What will be the value of the investment in 30 years?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q59872\">Show Solution<\/span><\/p>\n<div id=\"q59872\" class=\"hidden-answer\" style=\"display: none\">\n<p>$3,659,823.44<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=25526&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Calculating Continuous Decay<\/h3>\n<p>Radon-222 decays at a continuous rate of 17.3% per day. How much will 100 mg of Radon-222 decay to in 3 days?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q660828\">Show Solution<\/span><\/p>\n<div id=\"q660828\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since the substance is decaying, the rate, 17.3%, is negative. So, <em>r\u00a0<\/em>=\u00a0\u20130.173. The initial amount of radon-222 was 100 mg, so <em>a\u00a0<\/em>= 100. We use the continuous decay formula to find the value after <em>t\u00a0<\/em>= 3 days:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}A\\left(t\\right)\\hfill & =a{e}^{rt}\\hfill & \\text{Use the continuous growth formula}.\\hfill \\\\ \\hfill & =100{e}^{-0.173\\left(3\\right)} & \\text{Substitute known values for }a, r,\\text{ and }t.\\hfill \\\\ \\hfill & \\approx 59.5115\\hfill & \\text{Use a calculator to approximate}.\\hfill \\end{array}[\/latex]<\/p>\n<p>So 59.5115 mg of radon-222 will remain.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Using the data in the previous example, how much radon-222 will remain after one year?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q58534\">Show Solution<\/span><\/p>\n<div id=\"q58534\" class=\"hidden-answer\" style=\"display: none\">\n<p>3.77E-26 (This is calculator notation for the number written as [latex]3.77\\times {10}^{-26}[\/latex] in scientific notation. While the output of an exponential function is never zero, this number is so close to zero that for all practical purposes we can accept zero as the answer.)<\/p><\/div>\n<\/div>\n<\/div>\n<section id=\"fs-id1165135264762\" class=\"key-equations\"><\/section>\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-5337\">\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>Revision and Adaptation. <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>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 1495. <strong>Authored by<\/strong>: WebWork-Rochester. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 2453, 2942. <strong>Authored by<\/strong>: Anderson, Tophe. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 25526. <strong>Authored by<\/strong>: Morales, Lawrence. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-5337-1\">Todar, PhD, Kenneth. Todar's Online Textbook of Bacteriology. <a href=\"http:\/\/textbookofbacteriology.net\/growth_3.html\" target=\"_blank\" rel=\"noopener\">http:\/\/textbookofbacteriology.net\/growth_3.html<\/a>. <a href=\"#return-footnote-5337-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":167848,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question ID 1495\",\"author\":\"WebWork-Rochester\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 2453, 2942\",\"author\":\"Anderson, Tophe\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 25526\",\"author\":\"Morales, Lawrence\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5337","chapter","type-chapter","status-publish","hentry"],"part":5352,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/5337","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/wp-json\/wp\/v2\/users\/167848"}],"version-history":[{"count":24,"href":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/5337\/revisions"}],"predecessor-version":[{"id":5808,"href":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/5337\/revisions\/5808"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/5352"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/5337\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/wp-json\/wp\/v2\/media?parent=5337"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=5337"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/wp-json\/wp\/v2\/contributor?post=5337"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/wp-json\/wp\/v2\/license?post=5337"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}