{"id":15977,"date":"2019-12-05T05:17:11","date_gmt":"2019-12-05T05:17:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/chapter\/convert-from-logarithmic-to-exponential-form\/"},"modified":"2019-12-05T05:17:31","modified_gmt":"2019-12-05T05:17:31","slug":"convert-from-logarithmic-to-exponential-form","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/chapter\/convert-from-logarithmic-to-exponential-form\/","title":{"raw":"Define Logarithmic Functions","rendered":"Define Logarithmic Functions"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Convert from logarithmic to exponential form<\/li>\n \t<li>Convert from exponential to logarithmic form<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165135192781\">In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were&nbsp;[latex]500[\/latex] times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is [latex]{10}^{x}=500[\/latex], where [latex]x[\/latex] represents the difference in magnitudes on the <strong>Richter Scale<\/strong>. How would we solve for [latex]x[\/latex]?<\/p>\nWe have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve [latex]{10}^{x}=500[\/latex]. We know that [latex]{10}^{2}=100[\/latex] and [latex]{10}^{3}=1000[\/latex], so it is clear that [latex]x[\/latex]&nbsp;must be some value between&nbsp;[latex]2[\/latex] and&nbsp;[latex]3[\/latex], since [latex]y={10}^{x}[\/latex] is increasing. We can examine a graph&nbsp;to better estimate the solution.\n\n<img 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\">\n<p id=\"fs-id1165137662989\">Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph above&nbsp;passes the horizontal line test. The exponential function [latex]y={b}^{x}[\/latex] is <strong>one-to-one<\/strong>, so its inverse, [latex]x={b}^{y}[\/latex] is also a function. As is the case with all inverse functions, we simply interchange <em>x<\/em>&nbsp;and <em>y<\/em>&nbsp;and solve for <em>y<\/em>&nbsp;to find the inverse function. To represent <em>y<\/em>&nbsp;as a function of <em>x<\/em>, we use a logarithmic function of the form [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]. The base <em>b<\/em>&nbsp;<strong>logarithm<\/strong> of a number is the exponent by which we must raise <em>b<\/em>&nbsp;to get that number.<\/p>\n<p id=\"fs-id1165137404844\">We read a logarithmic expression as, \"The logarithm with base <em>b<\/em>&nbsp;of <em>x<\/em>&nbsp;is equal to <em>y<\/em>,\" or, simplified, \"log base <em>b<\/em>&nbsp;of <em>x<\/em>&nbsp;is <em>y<\/em>.\" We can also say, \"<em>b<\/em>&nbsp;raised to the power of <em>y<\/em>&nbsp;is <em>x<\/em>,\" because logs are exponents. For example, the base&nbsp;[latex]2[\/latex] logarithm of&nbsp;[latex]32[\/latex] is&nbsp;[latex]5[\/latex], because&nbsp;[latex]5[\/latex] is the exponent we must apply to&nbsp;[latex]2[\/latex] to get&nbsp;[latex]32[\/latex]. Since [latex]{2}^{5}=32[\/latex], we can write [latex]{\\mathrm{log}}_{2}32=5[\/latex]. We read this as \"log base&nbsp;[latex]2[\/latex] of&nbsp;[latex]32[\/latex] is&nbsp;[latex]5[\/latex].\"<\/p>\n<p id=\"fs-id1165137597501\">We can express the relationship between logarithmic form and its corresponding exponential form as follows:<\/p>\n\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>\n<p id=\"fs-id1165137678993\">Note that the base <em>b<\/em>&nbsp;is always positive.<span id=\"fs-id1165137696233\">\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><\/p>\n<p id=\"fs-id1165137400957\">Because logarithms are functions, they are most correctly written as [latex]{\\mathrm{log}}_{b}\\left(x\\right)[\/latex], using parentheses to denote function evaluation, just as we would with [latex]f\\left(x\\right)[\/latex]. However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as [latex]{\\mathrm{log}}_{b}x[\/latex]. Note that many calculators require parentheses around the <em>x<\/em>.<\/p>\n<p id=\"fs-id1165137827516\">We can illustrate the notation of logarithms as follows:<span id=\"fs-id1165137771679\">\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/05051928\/CNX_Precalc_Figure_04_03_0032.jpg\" alt=\"logb (c) = a means b to the A power equals C.\" width=\"487\" height=\"101\"><\/span><\/p>\n<p id=\"fs-id1165137575165\">Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] and [latex]y={b}^{x}[\/latex] are inverse functions.<\/p>\n\n<div id=\"fs-id1165137472937\" class=\"note textbox\">\n<h3 class=\"title\">Definition of the Logarithmic Function<\/h3>\n<p id=\"fs-id1165137704597\">A <strong>logarithm<\/strong> base <em>b<\/em>&nbsp;of a positive number <em>x<\/em>&nbsp;satisfies the following definition.<\/p>\n<p id=\"fs-id1165137584967\">For [latex]x&gt;0,b&gt;0,b\\ne 1[\/latex],<\/p>\n\n<div id=\"fs-id1165137433829\" class=\"equation\" style=\"text-align: center;\">[latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\text{ is the same as }{b}^{y}=x[\/latex]<\/div>\n<p id=\"fs-id1165137893373\">where<\/p>\n\n<ul id=\"fs-id1165135530561\">\n \t<li>we read [latex]{\\mathrm{log}}_{b}\\left(x\\right)[\/latex] as, \"the logarithm with base <em>b<\/em>&nbsp;of <em>x<\/em>\" or the \"log base <em>b<\/em>&nbsp;of <em>x<\/em>.\"<\/li>\n \t<li>the logarithm <em>y<\/em>&nbsp;is the exponent to which <em>b<\/em>&nbsp;must be raised to get <em>x<\/em>.<\/li>\n<\/ul>\n<p id=\"fs-id1165137547773\">Also, since the logarithmic and exponential functions switch the <em>x<\/em>&nbsp;and <em>y<\/em>&nbsp;values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,<\/p>\n\n<ul id=\"fs-id1165137643167\">\n \t<li>the domain of the logarithm function with base [latex]b \\text{ is} \\left(0,\\infty \\right)[\/latex].<\/li>\n \t<li>the range of the logarithm function with base [latex]b \\text{ is} \\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\n<\/ul>\n<\/div>\nIn our first example, we will convert logarithmic equations into exponential equations.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p id=\"fs-id1165137580570\">Write the following logarithmic equations in exponential form.<\/p>\n\n<ol id=\"fs-id1165137705346\">\n \t<li>[latex]{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}[\/latex]<\/li>\n \t<li>[latex]{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex]\n[reveal-answer q=\"161275\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"161275\"]\n<p id=\"fs-id1165137408172\">First, identify the values of <em>b<\/em>,&nbsp;<em>y<\/em>, and&nbsp;<em>x<\/em>. Then, write the equation in the form [latex]{b}^{y}=x[\/latex].<\/p>\n\n<ol id=\"fs-id1165137705659\">\n \t<li>[latex]{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}[\/latex]\n<p id=\"fs-id1165137602796\">Here, [latex]b=6,y=\\frac{1}{2},\\text{and } x=\\sqrt{6}[\/latex]. Therefore, the equation [latex]{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}[\/latex] can be written as [latex]{6}^{\\frac{1}{2}}=\\sqrt{6}[\/latex].<\/p>\n<\/li>\n \t<li>[latex]{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex]\n<p id=\"fs-id1165137698078\">Here, <em>b&nbsp;<\/em>=[latex]3[\/latex], <em>y&nbsp;<\/em>=[latex]2[\/latex], and <em>x&nbsp;<\/em>=[latex]9[\/latex]. Therefore, the equation [latex]{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex] can be written as [latex]{3}^{2}=9[\/latex].<\/p>\n<\/li>\n<\/ol>\n[\/hidden-answer]<\/li>\n<\/ol>\n<\/div>\nIn the following video, we present more examples of rewriting logarithmic equations as exponential equations.\n\nhttps:\/\/youtu.be\/q9_s0wqhIXU\n<div id=\"fs-id1165137874700\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137806301\">How To: Given an equation in logarithmic form [latex]{\\mathrm{log}}_{b}\\left(x\\right)=y[\/latex], convert it to exponential form<\/h3>\n<ol id=\"fs-id1165137641669\">\n \t<li>Examine the equation [latex]y={\\mathrm{log}}_{b}x[\/latex] and identify <em>b<\/em>, <em>y<\/em>, and <em>x<\/em>.<\/li>\n \t<li>Rewrite [latex]{\\mathrm{log}}_{b}x=y[\/latex] as [latex]{b}^{y}=x[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p id=\"eip-id1549475\">Can we take the logarithm of a negative number? Re-read the definition of a logarithm and formulate an answer. &nbsp;Think about the behavior of exponents. &nbsp;You can use the textbox below to formulate your ideas before you look at an answer.<\/p>\n[practice-area rows=\"1\"][\/practice-area]\n[reveal-answer q=\"162494\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"162494\"]\n<p id=\"fs-id1165137653864\">No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<h2>&nbsp;Convert from Exponential to Logarithmic Form<\/h2>\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].\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p id=\"fs-id1165137804412\">Write the following exponential equations in logarithmic form.<\/p>\n\n<ol id=\"fs-id1165135192287\">\n \t<li>[latex]{2}^{3}=8[\/latex]<\/li>\n \t<li>[latex]{5}^{2}=25[\/latex]<\/li>\n \t<li>[latex]{10}^{-4}=\\frac{1}{10,000}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"516026\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"516026\"]\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\n<ol id=\"fs-id1165137573458\">\n \t<li>[latex]{2}^{3}=8[\/latex]\n<p id=\"fs-id1165137466396\">Here, [latex]b=2[\/latex], [latex]x=3[\/latex], and&nbsp;&nbsp;[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 \t<li>[latex]{5}^{2}=25[\/latex]\n<p id=\"fs-id1165135193035\">Here,&nbsp;&nbsp;[latex]b=5[\/latex], [latex]x=2[\/latex], and&nbsp;&nbsp;[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 \t<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[\/hidden-answer]\n\n<\/div>\nIn our last video, we show more examples of writing logarithmic equations as exponential equations.\n\nhttps:\/\/youtu.be\/9_GPPUWEJQQ\n<h2>Summary<\/h2>\nThe base <em>b<\/em>&nbsp;<strong>logarithm<\/strong> of a number is the exponent by which we must raise <em>b<\/em>&nbsp;to get that number. Logarithmic functions are the inverse of exponential functions, and it is often easier to understand them through this lens.&nbsp;We can never take the logarithm of a negative number, therefore&nbsp;[latex]{\\mathrm{log}}_{b}\\left(x\\right)=y[\/latex] is defined for [latex]b&gt;0[\/latex]\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Convert from logarithmic to exponential form<\/li>\n<li>Convert from exponential to logarithmic form<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165135192781\">In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were&nbsp;[latex]500[\/latex] times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is [latex]{10}^{x}=500[\/latex], where [latex]x[\/latex] represents the difference in magnitudes on the <strong>Richter Scale<\/strong>. How would we solve for [latex]x[\/latex]?<\/p>\n<p>We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve [latex]{10}^{x}=500[\/latex]. We know that [latex]{10}^{2}=100[\/latex] and [latex]{10}^{3}=1000[\/latex], so it is clear that [latex]x[\/latex]&nbsp;must be some value between&nbsp;[latex]2[\/latex] and&nbsp;[latex]3[\/latex], since [latex]y={10}^{x}[\/latex] is increasing. We can examine a graph&nbsp;to better estimate the solution.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" 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 id=\"fs-id1165137662989\">Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph above&nbsp;passes the horizontal line test. The exponential function [latex]y={b}^{x}[\/latex] is <strong>one-to-one<\/strong>, so its inverse, [latex]x={b}^{y}[\/latex] is also a function. As is the case with all inverse functions, we simply interchange <em>x<\/em>&nbsp;and <em>y<\/em>&nbsp;and solve for <em>y<\/em>&nbsp;to find the inverse function. To represent <em>y<\/em>&nbsp;as a function of <em>x<\/em>, we use a logarithmic function of the form [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]. The base <em>b<\/em>&nbsp;<strong>logarithm<\/strong> of a number is the exponent by which we must raise <em>b<\/em>&nbsp;to get that number.<\/p>\n<p id=\"fs-id1165137404844\">We read a logarithmic expression as, &#8220;The logarithm with base <em>b<\/em>&nbsp;of <em>x<\/em>&nbsp;is equal to <em>y<\/em>,&#8221; or, simplified, &#8220;log base <em>b<\/em>&nbsp;of <em>x<\/em>&nbsp;is <em>y<\/em>.&#8221; We can also say, &#8220;<em>b<\/em>&nbsp;raised to the power of <em>y<\/em>&nbsp;is <em>x<\/em>,&#8221; because logs are exponents. For example, the base&nbsp;[latex]2[\/latex] logarithm of&nbsp;[latex]32[\/latex] is&nbsp;[latex]5[\/latex], because&nbsp;[latex]5[\/latex] is the exponent we must apply to&nbsp;[latex]2[\/latex] to get&nbsp;[latex]32[\/latex]. Since [latex]{2}^{5}=32[\/latex], we can write [latex]{\\mathrm{log}}_{2}32=5[\/latex]. We read this as &#8220;log base&nbsp;[latex]2[\/latex] of&nbsp;[latex]32[\/latex] is&nbsp;[latex]5[\/latex].&#8221;<\/p>\n<p id=\"fs-id1165137597501\">We can express the relationship between logarithmic form and its corresponding exponential form 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<p id=\"fs-id1165137678993\">Note that the base <em>b<\/em>&nbsp;is always positive.<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<p id=\"fs-id1165137400957\">Because logarithms are functions, they are most correctly written as [latex]{\\mathrm{log}}_{b}\\left(x\\right)[\/latex], using parentheses to denote function evaluation, just as we would with [latex]f\\left(x\\right)[\/latex]. However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as [latex]{\\mathrm{log}}_{b}x[\/latex]. Note that many calculators require parentheses around the <em>x<\/em>.<\/p>\n<p id=\"fs-id1165137827516\">We can illustrate the notation of logarithms as follows:<span id=\"fs-id1165137771679\"><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\/05051928\/CNX_Precalc_Figure_04_03_0032.jpg\" alt=\"logb (c) = a means b to the A power equals C.\" width=\"487\" height=\"101\" \/><\/span><\/p>\n<p id=\"fs-id1165137575165\">Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] and [latex]y={b}^{x}[\/latex] are inverse functions.<\/p>\n<div id=\"fs-id1165137472937\" class=\"note textbox\">\n<h3 class=\"title\">Definition of the Logarithmic Function<\/h3>\n<p id=\"fs-id1165137704597\">A <strong>logarithm<\/strong> base <em>b<\/em>&nbsp;of a positive number <em>x<\/em>&nbsp;satisfies the following definition.<\/p>\n<p id=\"fs-id1165137584967\">For [latex]x>0,b>0,b\\ne 1[\/latex],<\/p>\n<div id=\"fs-id1165137433829\" class=\"equation\" style=\"text-align: center;\">[latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\text{ is the same as }{b}^{y}=x[\/latex]<\/div>\n<p id=\"fs-id1165137893373\">where<\/p>\n<ul id=\"fs-id1165135530561\">\n<li>we read [latex]{\\mathrm{log}}_{b}\\left(x\\right)[\/latex] as, &#8220;the logarithm with base <em>b<\/em>&nbsp;of <em>x<\/em>&#8221; or the &#8220;log base <em>b<\/em>&nbsp;of <em>x<\/em>.&#8221;<\/li>\n<li>the logarithm <em>y<\/em>&nbsp;is the exponent to which <em>b<\/em>&nbsp;must be raised to get <em>x<\/em>.<\/li>\n<\/ul>\n<p id=\"fs-id1165137547773\">Also, since the logarithmic and exponential functions switch the <em>x<\/em>&nbsp;and <em>y<\/em>&nbsp;values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,<\/p>\n<ul id=\"fs-id1165137643167\">\n<li>the domain of the logarithm function with base [latex]b \\text{ is} \\left(0,\\infty \\right)[\/latex].<\/li>\n<li>the range of the logarithm function with base [latex]b \\text{ is} \\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\n<\/ul>\n<\/div>\n<p>In our first example, we will convert logarithmic equations into exponential equations.<\/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 id=\"fs-id1165137705346\">\n<li>[latex]{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}[\/latex]<\/li>\n<li>[latex]{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex]\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q161275\">Show Solution<\/span><\/p>\n<div id=\"q161275\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137408172\">First, identify the values of <em>b<\/em>,&nbsp;<em>y<\/em>, and&nbsp;<em>x<\/em>. Then, write the equation in the form [latex]{b}^{y}=x[\/latex].<\/p>\n<ol id=\"fs-id1165137705659\">\n<li>[latex]{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}[\/latex]\n<p id=\"fs-id1165137602796\">Here, [latex]b=6,y=\\frac{1}{2},\\text{and } x=\\sqrt{6}[\/latex]. Therefore, the equation [latex]{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}[\/latex] can be written as [latex]{6}^{\\frac{1}{2}}=\\sqrt{6}[\/latex].<\/p>\n<\/li>\n<li>[latex]{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex]\n<p id=\"fs-id1165137698078\">Here, <em>b&nbsp;<\/em>=[latex]3[\/latex], <em>y&nbsp;<\/em>=[latex]2[\/latex], and <em>x&nbsp;<\/em>=[latex]9[\/latex]. Therefore, the equation [latex]{\\mathrm{log}}_{3}\\left(9\\right)=2[\/latex] can be written as [latex]{3}^{2}=9[\/latex].<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<p>In the following video, we present 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<div id=\"fs-id1165137874700\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137806301\">How To: Given an equation in logarithmic form [latex]{\\mathrm{log}}_{b}\\left(x\\right)=y[\/latex], convert it to exponential form<\/h3>\n<ol id=\"fs-id1165137641669\">\n<li>Examine the equation [latex]y={\\mathrm{log}}_{b}x[\/latex] and identify <em>b<\/em>, <em>y<\/em>, and <em>x<\/em>.<\/li>\n<li>Rewrite [latex]{\\mathrm{log}}_{b}x=y[\/latex] as [latex]{b}^{y}=x[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p id=\"eip-id1549475\">Can we take the logarithm of a negative number? Re-read the definition of a logarithm and formulate an answer. &nbsp;Think about the behavior of exponents. &nbsp;You can use the textbox below to formulate your ideas before you look at an answer.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q162494\">Show Solution<\/span><\/p>\n<div id=\"q162494\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137653864\">No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>&nbsp;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 id=\"fs-id1165135192287\">\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=\"q516026\">Show Solution<\/span><\/p>\n<div id=\"q516026\" 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 id=\"fs-id1165137573458\">\n<li>[latex]{2}^{3}=8[\/latex]\n<p id=\"fs-id1165137466396\">Here, [latex]b=2[\/latex], [latex]x=3[\/latex], and&nbsp;&nbsp;[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,&nbsp;&nbsp;[latex]b=5[\/latex], [latex]x=2[\/latex], and&nbsp;&nbsp;[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>In our last video, we show more examples of writing logarithmic equations as exponential equations.<\/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<h2>Summary<\/h2>\n<p>The base <em>b<\/em>&nbsp;<strong>logarithm<\/strong> of a number is the exponent by which we must raise <em>b<\/em>&nbsp;to get that number. Logarithmic functions are the inverse of exponential functions, and it is often easier to understand them through this lens.&nbsp;We can never take the logarithm of a negative number, therefore&nbsp;[latex]{\\mathrm{log}}_{b}\\left(x\\right)=y[\/latex] is defined for [latex]b>0[\/latex]<\/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-15977\">\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=\"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>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><\/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":23485,"menu_order":9,"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 : 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