{"id":1438,"date":"2021-08-24T18:15:48","date_gmt":"2021-08-24T18:15:48","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1438"},"modified":"2022-01-31T21:55:08","modified_gmt":"2022-01-31T21:55:08","slug":"logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/logarithmic-functions\/","title":{"raw":"Logarithmic Functions","rendered":"Logarithmic Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Convert from logarithmic form to exponential form<\/li>\r\n \t<li>Convert from exponential form to logarithmic form<\/li>\r\n \t<li>Evaluate functions involving natural logarithms<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165135192781\">In the graph of the exponential function [latex]y=10^{x}[\/latex] shown below, we can see that every positive value of [latex]y[\/latex] corresponds to exactly one value of [latex]x[\/latex]. This is the\u00a0<strong>one-to-one<\/strong> property of exponential functions. For any positive number [latex]b \\neq 1[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]b^{x}=b^{y}[\/latex] if and only if [latex]x=y[\/latex].<\/p>\r\nWe may want to find what value of [latex]x[\/latex] produces a value of [latex]y = 500[\/latex]. We know that [latex]10^{2}=100[\/latex] and [latex]10^{3}=1,000[\/latex], so the correct value of [latex]x[\/latex] is somewhere between [latex]2[\/latex] and [latex]3[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051924\/CNX_Precalc_Figure_04_03_0022.jpg\" alt=\"Graph of the intersections of the equations y=10^x and y=500.\" width=\"487\" height=\"477\" \/>\r\n\r\nLogarithmic functions are used to solve problems such as this.\r\n\r\nWe define the <strong>logarithmic function with base b<\/strong> as follows:\r\n<div id=\"eip-604\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left(x\\right)=y\\Leftrightarrow {b}^{y}=x,\\text{}b&gt;0,b\\ne 1[\/latex].<\/div>\r\n<div><\/div>\r\n<div>The two statements [latex]y=\\mathrm{log}_{b} x[\/latex] and [latex]b^{y}=x[\/latex] are equivalent. We say that [latex]y=\\mathrm{log}_{b} x[\/latex] is the logarithmic form of [latex]b^{y}=x[\/latex] is the exponential form.<\/div>\r\n<p id=\"fs-id1165137678993\">Note that the base <em>b<\/em>\u00a0is always positive.<\/p>\r\n<span id=\"fs-id1165137696233\">The value [latex]x[\/latex] in parentheses is called the argument. The argument is the quantity we are taking the logarithm of. Since [latex]b^{y}&gt;0[\/latex], and [latex]x=b^{y}[\/latex], the argument of a logarithm must be positive.<\/span>\r\n\r\n<span id=\"fs-id1165137696233\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051926\/CNX_Precalc_Figure_04_03_0042.jpg\" alt=\"Think b to the y equals x.\" width=\"487\" height=\"83\" \/><\/span>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p id=\"fs-id1165137580570\">Write the following logarithmic equations in exponential form.<\/p>\r\n\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\mathrm{log}_{2} 8 =3[\/latex]<\/li>\r\n \t<li>[latex]\\mathrm{log}_{3} 9 =2[\/latex]<\/li>\r\n \t<li>[latex]\\mathrm{log}_{4} \\frac{1}{16} =-2[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"333093\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"333093\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\mathrm{log}_{2} 8 =3[\/latex] has base [latex]b=2[\/latex], argument [latex]x=8[\/latex] and exponent [latex]y=3[\/latex]. Writing this expression in exponential form we have [latex]2^{3}=8[\/latex].<\/li>\r\n \t<li>[latex]\\mathrm{log}_{3} 9 =2[\/latex] has base [latex]b=3[\/latex], argument [latex]x=9[\/latex], and exponent [latex]y=2[\/latex]. Writing this expression in exponential form we have [latex]3^{2}=9[\/latex].<\/li>\r\n \t<li>[latex]\\mathrm{log}_{4} \\frac{1}{16} =-2[\/latex] has [latex]b=4[\/latex], argument [latex]x=\\frac{1}{16}[\/latex], and exponent [latex]y=-2[\/latex]. Writing this expression in exponential form we have [latex]4^{-2}=\\frac{1}{16}[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video are more examples of rewriting logarithmic equations as exponential equations.\r\n\r\nhttps:\/\/youtu.be\/q9_s0wqhIXU\r\n<h2>Convert from Exponential to Logarithmic Form<\/h2>\r\nTo convert from exponential form to logarithmic form, we follow the same steps in reverse. We identify the base <em>b<\/em>, exponent <em>x<\/em>, and output <em>y<\/em>. Then we write [latex]x={\\mathrm{log}}_{b}\\left(y\\right)[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p id=\"fs-id1165137804412\">Write the following exponential equations in logarithmic form.<\/p>\r\n\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]{2}^{3}=8[\/latex]<\/li>\r\n \t<li>[latex]{5}^{2}=25[\/latex]<\/li>\r\n \t<li>[latex]{10}^{-4}=\\frac{1}{10,000}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"213399\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"213399\"]\r\n<p id=\"fs-id1165137474116\">First, identify the values of <em>b<\/em>, <em>y<\/em>, and <em>x<\/em>. Then, write the equation in the form [latex]x={\\mathrm{log}}_{b}\\left(y\\right)[\/latex].<\/p>\r\n\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]{2}^{3}=8[\/latex]\r\n<p id=\"fs-id1165137466396\">Here, [latex]b=2[\/latex], [latex]x=3[\/latex], and\u00a0\u00a0[latex]y=8[\/latex]. Therefore, the equation [latex]{2}^{3}=8[\/latex] can be written as [latex]{\\mathrm{log}}_{2}\\left(8\\right)=3[\/latex].<\/p>\r\n<\/li>\r\n \t<li>[latex]{5}^{2}=25[\/latex]\r\n<p id=\"fs-id1165135193035\">Here,\u00a0\u00a0[latex]b=5[\/latex], [latex]x=2[\/latex], and\u00a0\u00a0[latex]y=25[\/latex]. Therefore, the equation [latex]{5}^{2}=25[\/latex] can be written as [latex]{\\mathrm{log}}_{5}\\left(25\\right)=2[\/latex].<\/p>\r\n<\/li>\r\n \t<li>[latex]{10}^{-4}=\\frac{1}{10,000}[\/latex]\r\n<p id=\"fs-id1165135187822\">Here, [latex]b=10[\/latex], [latex]x=\u20134[\/latex], and [latex]y=\\frac{1}{10,000}[\/latex]. Therefore, the equation [latex]{10}^{-4}=\\frac{1}{10,000}[\/latex] can be written as [latex]{\\text{log}}_{10}\\left(\\frac{1}{10,000}\\right)=-4[\/latex].<\/p>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe next video shows more examples of writing exponential expressions in logarithmic form.\r\n\r\nhttps:\/\/youtu.be\/9_GPPUWEJQQ\r\n\r\nMany calculators can evaluate only two types of logarithmic functions.\r\n\r\n<strong>Common logarithms<\/strong>, written [latex]\\mathrm{log}(x)[\/latex], have an implied base [latex]b=10[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\mathrm{log}(x) = \\mathrm{log}_{10} (x)[\/latex]<\/p>\r\n<strong>Natural logarithms<\/strong>, written [latex]\\mathrm{ln}(x)[\/latex], have base [latex]e[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\mathrm{ln}(x) = \\mathrm{log}_{e} (x)[\/latex]<\/p>\r\nWe can use logarithms to solve exponential equations. That is, equations where the variable appears in an exponent.\r\n\r\nConsider the example at the beginning of this section. We wanted to find the value of [latex]x[\/latex] such that [latex]10^{x}=500[\/latex]. If we rewrite this exponential expression in logarithmic form we have [latex]\\mathrm{log}_{10} 500=x[\/latex] or simply [latex]\\mathrm{log} \\ 500=x[\/latex]. Using a calculator we can find an approximate value for [latex]x, x \\approx 2.69897[\/latex].\r\n\r\nThe most frequently used base for logarithms is [latex]e[\/latex]. In the next example, we will evaluate a natural logarithm using a calculator.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nEvaluate [latex]y=\\mathrm{ln}\\left(500\\right)[\/latex] to four decimal places using a calculator.\r\n[reveal-answer q=\"957920\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"957920\"]\r\n<ul id=\"fs-id1165137563770\">\r\n \t<li>Press <strong>[LN]<\/strong>.<\/li>\r\n \t<li>Enter\u00a0[latex]500[\/latex], followed by <strong>[ ) ]<\/strong>.<\/li>\r\n \t<li>Press <strong>[ENTER]<\/strong>.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137645024\">Rounding to four decimal places, [latex]\\mathrm{ln}\\left(500\\right)\\approx 6.2146[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn our next video, we show more examples of how to evaluate natural logarithms using a calculator.\r\n\r\nhttps:\/\/youtu.be\/Rpounu3epSc","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Convert from logarithmic form to exponential form<\/li>\n<li>Convert from exponential form to logarithmic form<\/li>\n<li>Evaluate functions involving natural logarithms<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165135192781\">In the graph of the exponential function [latex]y=10^{x}[\/latex] shown below, we can see that every positive value of [latex]y[\/latex] corresponds to exactly one value of [latex]x[\/latex]. This is the\u00a0<strong>one-to-one<\/strong> property of exponential functions. For any positive number [latex]b \\neq 1[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]b^{x}=b^{y}[\/latex] if and only if [latex]x=y[\/latex].<\/p>\n<p>We may want to find what value of [latex]x[\/latex] produces a value of [latex]y = 500[\/latex]. We know that [latex]10^{2}=100[\/latex] and [latex]10^{3}=1,000[\/latex], so the correct value of [latex]x[\/latex] is somewhere between [latex]2[\/latex] and [latex]3[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051924\/CNX_Precalc_Figure_04_03_0022.jpg\" alt=\"Graph of the intersections of the equations y=10^x and y=500.\" width=\"487\" height=\"477\" \/><\/p>\n<p>Logarithmic functions are used to solve problems such as this.<\/p>\n<p>We define the <strong>logarithmic function with base b<\/strong> as follows:<\/p>\n<div id=\"eip-604\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left(x\\right)=y\\Leftrightarrow {b}^{y}=x,\\text{}b>0,b\\ne 1[\/latex].<\/div>\n<div><\/div>\n<div>The two statements [latex]y=\\mathrm{log}_{b} x[\/latex] and [latex]b^{y}=x[\/latex] are equivalent. We say that [latex]y=\\mathrm{log}_{b} x[\/latex] is the logarithmic form of [latex]b^{y}=x[\/latex] is the exponential form.<\/div>\n<p id=\"fs-id1165137678993\">Note that the base <em>b<\/em>\u00a0is always positive.<\/p>\n<p><span id=\"fs-id1165137696233\">The value [latex]x[\/latex] in parentheses is called the argument. The argument is the quantity we are taking the logarithm of. Since [latex]b^{y}>0[\/latex], and [latex]x=b^{y}[\/latex], the argument of a logarithm must be positive.<\/span><\/p>\n<p><span id=\"fs-id1165137696233\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051926\/CNX_Precalc_Figure_04_03_0042.jpg\" alt=\"Think b to the y equals x.\" width=\"487\" height=\"83\" \/><\/span><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p id=\"fs-id1165137580570\">Write the following logarithmic equations in exponential form.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\mathrm{log}_{2} 8 =3[\/latex]<\/li>\n<li>[latex]\\mathrm{log}_{3} 9 =2[\/latex]<\/li>\n<li>[latex]\\mathrm{log}_{4} \\frac{1}{16} =-2[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q333093\">Show Answer<\/span><\/p>\n<div id=\"q333093\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\mathrm{log}_{2} 8 =3[\/latex] has base [latex]b=2[\/latex], argument [latex]x=8[\/latex] and exponent [latex]y=3[\/latex]. Writing this expression in exponential form we have [latex]2^{3}=8[\/latex].<\/li>\n<li>[latex]\\mathrm{log}_{3} 9 =2[\/latex] has base [latex]b=3[\/latex], argument [latex]x=9[\/latex], and exponent [latex]y=2[\/latex]. Writing this expression in exponential form we have [latex]3^{2}=9[\/latex].<\/li>\n<li>[latex]\\mathrm{log}_{4} \\frac{1}{16} =-2[\/latex] has [latex]b=4[\/latex], argument [latex]x=\\frac{1}{16}[\/latex], and exponent [latex]y=-2[\/latex]. Writing this expression in exponential form we have [latex]4^{-2}=\\frac{1}{16}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video are more examples of rewriting logarithmic equations as exponential equations.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Write Logarithmic Equations as Exponential Equations\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/q9_s0wqhIXU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Convert from Exponential to Logarithmic Form<\/h2>\n<p>To convert from exponential form to logarithmic form, we follow the same steps in reverse. We identify the base <em>b<\/em>, exponent <em>x<\/em>, and output <em>y<\/em>. Then we write [latex]x={\\mathrm{log}}_{b}\\left(y\\right)[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p id=\"fs-id1165137804412\">Write the following exponential equations in logarithmic form.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]{2}^{3}=8[\/latex]<\/li>\n<li>[latex]{5}^{2}=25[\/latex]<\/li>\n<li>[latex]{10}^{-4}=\\frac{1}{10,000}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q213399\">Show Answer<\/span><\/p>\n<div id=\"q213399\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137474116\">First, identify the values of <em>b<\/em>, <em>y<\/em>, and <em>x<\/em>. Then, write the equation in the form [latex]x={\\mathrm{log}}_{b}\\left(y\\right)[\/latex].<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]{2}^{3}=8[\/latex]\n<p id=\"fs-id1165137466396\">Here, [latex]b=2[\/latex], [latex]x=3[\/latex], and\u00a0\u00a0[latex]y=8[\/latex]. Therefore, the equation [latex]{2}^{3}=8[\/latex] can be written as [latex]{\\mathrm{log}}_{2}\\left(8\\right)=3[\/latex].<\/p>\n<\/li>\n<li>[latex]{5}^{2}=25[\/latex]\n<p id=\"fs-id1165135193035\">Here,\u00a0\u00a0[latex]b=5[\/latex], [latex]x=2[\/latex], and\u00a0\u00a0[latex]y=25[\/latex]. Therefore, the equation [latex]{5}^{2}=25[\/latex] can be written as [latex]{\\mathrm{log}}_{5}\\left(25\\right)=2[\/latex].<\/p>\n<\/li>\n<li>[latex]{10}^{-4}=\\frac{1}{10,000}[\/latex]\n<p id=\"fs-id1165135187822\">Here, [latex]b=10[\/latex], [latex]x=\u20134[\/latex], and [latex]y=\\frac{1}{10,000}[\/latex]. Therefore, the equation [latex]{10}^{-4}=\\frac{1}{10,000}[\/latex] can be written as [latex]{\\text{log}}_{10}\\left(\\frac{1}{10,000}\\right)=-4[\/latex].<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>The next video shows more examples of writing exponential expressions in logarithmic form.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Write Exponential Equations as Logarithmic Equations\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/9_GPPUWEJQQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Many calculators can evaluate only two types of logarithmic functions.<\/p>\n<p><strong>Common logarithms<\/strong>, written [latex]\\mathrm{log}(x)[\/latex], have an implied base [latex]b=10[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\mathrm{log}(x) = \\mathrm{log}_{10} (x)[\/latex]<\/p>\n<p><strong>Natural logarithms<\/strong>, written [latex]\\mathrm{ln}(x)[\/latex], have base [latex]e[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\mathrm{ln}(x) = \\mathrm{log}_{e} (x)[\/latex]<\/p>\n<p>We can use logarithms to solve exponential equations. That is, equations where the variable appears in an exponent.<\/p>\n<p>Consider the example at the beginning of this section. We wanted to find the value of [latex]x[\/latex] such that [latex]10^{x}=500[\/latex]. If we rewrite this exponential expression in logarithmic form we have [latex]\\mathrm{log}_{10} 500=x[\/latex] or simply [latex]\\mathrm{log} \\ 500=x[\/latex]. Using a calculator we can find an approximate value for [latex]x, x \\approx 2.69897[\/latex].<\/p>\n<p>The most frequently used base for logarithms is [latex]e[\/latex]. In the next example, we will evaluate a natural logarithm using a calculator.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]y=\\mathrm{ln}\\left(500\\right)[\/latex] to four decimal places using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q957920\">Show Solution<\/span><\/p>\n<div id=\"q957920\" class=\"hidden-answer\" style=\"display: none\">\n<ul id=\"fs-id1165137563770\">\n<li>Press <strong>[LN]<\/strong>.<\/li>\n<li>Enter\u00a0[latex]500[\/latex], followed by <strong>[ ) ]<\/strong>.<\/li>\n<li>Press <strong>[ENTER]<\/strong>.<\/li>\n<\/ul>\n<p id=\"fs-id1165137645024\">Rounding to four decimal places, [latex]\\mathrm{ln}\\left(500\\right)\\approx 6.2146[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In our next video, we show more examples of how to evaluate natural logarithms using a calculator.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Evaluate Natural Logarithms on the Calculator\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Rpounu3epSc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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-1438\">\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>Ex: Write Exponential Equations as Logarithmic Equations. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/9_GPPUWEJQQ\">https:\/\/youtu.be\/9_GPPUWEJQQ<\/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>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=\"https:\/\/openstax.org\/books\/precalculus\/pages\/1-introduction-to-functions\">https:\/\/openstax.org\/books\/precalculus\/pages\/1-introduction-to-functions<\/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>: Access for free at https:\/\/openstax.org\/books\/precalculus\/pages\/1-introduction-to-functions<\/li><li>Ex: Write Logarithmic Equations as Exponential Equations. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/q9_s0wqhIXU\">https:\/\/youtu.be\/q9_s0wqhIXU<\/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>Ex: Evaluate Natural Logarithms on the Calculator. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Rpounu3epSc\">https:\/\/youtu.be\/Rpounu3epSc<\/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 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