{"id":268,"date":"2023-06-21T13:22:50","date_gmt":"2023-06-21T13:22:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/evaluate-logarithms\/"},"modified":"2023-07-04T04:18:53","modified_gmt":"2023-07-04T04:18:53","slug":"evaluate-logarithms","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/evaluate-logarithms\/","title":{"raw":"\u25aa   Evaluating Logarithms","rendered":"\u25aa   Evaluating Logarithms"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Evaluate logarithms with and without a calculator.<\/li>\r\n \t<li>Evaluate logarithms with base 10 and base e.<\/li>\r\n<\/ul>\r\n<\/div>\r\nKnowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider [latex]{\\mathrm{log}}_{2}8[\/latex]. We ask, \"To what exponent must 2\u00a0be raised in order to get 8?\" Because we already know [latex]{2}^{3}=8[\/latex], it follows that [latex]{\\mathrm{log}}_{2}8=3[\/latex].\r\n\r\nNow consider solving [latex]{\\mathrm{log}}_{7}49[\/latex] and [latex]{\\mathrm{log}}_{3}27[\/latex] mentally.\r\n<ul>\r\n \t<li>We ask, \"To what exponent must 7 be raised in order to get 49?\" We know [latex]{7}^{2}=49[\/latex]. Therefore, [latex]{\\mathrm{log}}_{7}49=2[\/latex].<\/li>\r\n \t<li>We ask, \"To what exponent must 3 be raised in order to get 27?\" We know [latex]{3}^{3}=27[\/latex]. Therefore, [latex]{\\mathrm{log}}_{3}27=3[\/latex].<\/li>\r\n<\/ul>\r\nEven some seemingly more complicated logarithms can be evaluated without a calculator. For example, let\u2019s evaluate [latex]{\\mathrm{log}}_{\\frac{2}{3}}\\frac{4}{9}[\/latex] mentally.\r\n<ul>\r\n \t<li>We ask, \"To what exponent must [latex]\\frac{2}{3}[\/latex] be raised in order to get [latex]\\frac{4}{9}[\/latex]? \" We know [latex]{2}^{2}=4[\/latex] and [latex]{3}^{2}=9[\/latex], so [latex]{\\left(\\frac{2}{3}\\right)}^{2}=\\frac{4}{9}[\/latex]. Therefore, [latex]{\\mathrm{log}}_{\\frac{2}{3}}\\left(\\frac{4}{9}\\right)=2[\/latex].<\/li>\r\n<\/ul>\r\n<div class=\"textbox examples\">\r\n<h3>tip for success<\/h3>\r\nIt may be tempting to use your calculator to evaluate these logarithms but try to evaluate them mentally as it will aid your understanding of what a logarithm is, and will help you navigate more complicated situations.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a logarithm of the form [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], evaluate it mentally<\/h3>\r\n<ol>\r\n \t<li>Rewrite the argument <em>x<\/em>\u00a0as a power of <em>b<\/em>: [latex]{b}^{y}=x[\/latex].<\/li>\r\n \t<li>Use previous knowledge of powers of <em>b<\/em>\u00a0to identify <em>y<\/em>\u00a0by asking, \"To what exponent should <em>b<\/em>\u00a0be raised in order to get <em>x<\/em>?\"<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving Logarithms Mentally<\/h3>\r\nSolve [latex]y={\\mathrm{log}}_{4}\\left(64\\right)[\/latex] without using a calculator.\r\n\r\n[reveal-answer q=\"879580\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"879580\"]\r\n\r\nFirst we rewrite the logarithm in exponential form: [latex]{4}^{y}=64[\/latex]. Next, we ask, \"To what exponent must 4 be raised in order to get 64?\"\r\n\r\nWe know [latex]{4}^{3}=64[\/latex]\r\n\r\nTherefore,\r\n\r\n[latex]\\mathrm{log}{}_{4}\\left(64\\right)=3[\/latex]\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\nSolve [latex]y={\\mathrm{log}}_{121}\\left(11\\right)[\/latex] without using a calculator.\r\n\r\n[reveal-answer q=\"143125\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"143125\"]\r\n\r\n[latex]{\\mathrm{log}}_{121}\\left(11\\right)=\\frac{1}{2}[\/latex] (recall that [latex]\\sqrt{121}={\\left(121\\right)}^{\\frac{1}{2}}=11[\/latex] )[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating the Logarithm of a Reciprocal<\/h3>\r\nEvaluate [latex]y={\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)[\/latex] without using a calculator.\r\n\r\n[reveal-answer q=\"861965\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"861965\"]\r\n\r\nFirst we rewrite the logarithm in exponential form: [latex]{3}^{y}=\\frac{1}{27}[\/latex]. Next, we ask, \"To what exponent must 3 be raised in order to get [latex]\\frac{1}{27}[\/latex]\"?\r\n\r\nWe know [latex]{3}^{3}=27[\/latex], but what must we do to get the reciprocal, [latex]\\frac{1}{27}[\/latex]? Recall from working with exponents that [latex]{b}^{-a}=\\frac{1}{{b}^{a}}[\/latex]. We use this information to write\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{3}^{-3}=\\frac{1}{{3}^{3}}=\\frac{1}{27}\\hfill \\end{array}[\/latex]<\/p>\r\nTherefore, [latex]{\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)=-3[\/latex].\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\nEvaluate [latex]y={\\mathrm{log}}_{2}\\left(\\frac{1}{32}\\right)[\/latex] without using a calculator.\r\n\r\n[reveal-answer q=\"765423\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"765423\"][latex]{\\mathrm{log}}_{2}\\left(\\frac{1}{32}\\right)=-5[\/latex][\/hidden-answer]\r\n\r\n[ohm_question]15905[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Using Common Logarithms<\/h2>\r\nSometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expressions [latex]\\text{log}(x)[\/latex] means [latex]\\text{log}_{10}(x).[\/latex] We call a base-[latex]10[\/latex] logarithm a\u00a0<strong>common logarithm.\u00a0<\/strong>Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Definition of the Common Logarithm<\/h3>\r\nA <strong>common logarithm<\/strong> is a logarithm with base [latex]10[\/latex]. We write [latex]\\text{log}_{10}(x).[\/latex] simply as [latex]\\text{log}(x)[\/latex]. The common logarithm of a positive number [latex]x[\/latex]\u00a0satisfies the following definition:\r\n\r\nFor [latex]x&gt;0[\/latex],\r\n<p style=\"text-align: center;\">[latex]y=\\text{log}(x)\\text{ is equivalent to }{10}^{y}=x[\/latex]<\/p>\r\n&nbsp;\r\n<p style=\"text-align: left;\">We read [latex]\\text{log}(x)[\/latex] as, \"the logarithm with base [latex]10[\/latex]\u00a0of [latex]x[\/latex]\" or \"the common logarithm of [latex]x[\/latex].\"<\/p>\r\nThe logarithm [latex]y[\/latex] is the exponent to which [latex]10[\/latex] must be raised to get [latex]x[\/latex].\r\n\r\nSince the functions [latex]y=10^{x}[\/latex] and [latex]y=\\mathrm{log}\\left(x\\right)[\/latex] are inverse functions, [latex]\\mathrm{log}\\left({10}^{x}\\right)=x[\/latex] for all [latex]x[\/latex]\u00a0and [latex]10^{\\mathrm{log}\\left(x\\right)}=x[\/latex] for [latex]x&gt;0[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a common logarithm Of the form [latex]y=\\mathrm{log}\\left(x\\right)[\/latex], evaluate it using a calculator+<\/h3>\r\n<ol>\r\n \t<li>Press <strong>[LOG]<\/strong>.<\/li>\r\n \t<li>Enter the value given for <em>x<\/em>, followed by <strong>[ ) ]<\/strong>.<\/li>\r\n \t<li>Press <strong>[ENTER]<\/strong>.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h2>Using Natural Logarithms<\/h2>\r\nThe most frequently used base for logarithms is <em>e<\/em>. Base <em>e<\/em>\u00a0logarithms are important in calculus and some scientific applications; they are called <strong>natural logarithms<\/strong>. The base <em>e<\/em>\u00a0logarithm, [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex], has its own notation, [latex]\\mathrm{ln}\\left(x\\right)[\/latex].\r\n\r\nMost values of [latex]\\mathrm{ln}\\left(x\\right)[\/latex] can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, [latex]\\mathrm{ln}1=0[\/latex]. For other natural logarithms, we can use the [latex]\\mathrm{ln}[\/latex] key that can be found on most scientific calculators. We can also find the natural logarithm of any power of <em>e<\/em>\u00a0using the inverse property of logarithms.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Definition of the Natural Logarithm<\/h3>\r\nA <strong>natural logarithm<\/strong> is a logarithm with base <em>e<\/em>. We write [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex] simply as [latex]\\mathrm{ln}\\left(x\\right)[\/latex]. The natural logarithm of a positive number <em>x<\/em>\u00a0satisfies the following definition:\r\n\r\nFor [latex]x&gt;0[\/latex],\r\n<p style=\"text-align: center;\">[latex]y=\\mathrm{ln}\\left(x\\right)\\text{ is equivalent to }{e}^{y}=x[\/latex]<\/p>\r\n&nbsp;\r\n\r\nWe read [latex]\\mathrm{ln}\\left(x\\right)[\/latex] as, \"the logarithm with base <em>e<\/em>\u00a0of <em>x<\/em>\" or \"the natural logarithm of <em>x<\/em>.\"\r\n\r\nThe logarithm [latex]y[\/latex]\u00a0is the exponent to which <em>e<\/em>\u00a0must be raised to get [latex]x[\/latex].\r\n\r\nSince the functions [latex]y=e^{x}[\/latex] and [latex]y=\\mathrm{ln}\\left(x\\right)[\/latex] are inverse functions, [latex]\\mathrm{ln}\\left({e}^{x}\\right)=x[\/latex] for all <em>x<\/em>\u00a0and [latex]e^{\\mathrm{ln}\\left(x\\right)}=x[\/latex] for [latex]x&gt;0[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a natural logarithm Of the form [latex]y=\\mathrm{ln}\\left(x\\right)[\/latex], evaluate it using a calculator+<\/h3>\r\n<ol>\r\n \t<li>Press <strong>[LN]<\/strong>.<\/li>\r\n \t<li>Enter the value given for <em>x<\/em>, followed by <strong>[ ) ]<\/strong>.<\/li>\r\n \t<li>Press <strong>[ENTER]<\/strong>.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating a Natural Logarithm Using a Calculator<\/h3>\r\nEvaluate [latex]y=\\mathrm{ln}\\left(500\\right)[\/latex] to four decimal places using a calculator.\r\n\r\n[reveal-answer q=\"810207\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"810207\"]\r\n<ul>\r\n \t<li>Press <strong>[LN]<\/strong>.<\/li>\r\n \t<li>Enter 500, followed by <strong>[ ) ]<\/strong>.<\/li>\r\n \t<li>Press <strong>[ENTER]<\/strong>.<\/li>\r\n<\/ul>\r\nRounding to four decimal places, [latex]\\mathrm{ln}\\left(500\\right)\\approx 6.2146[\/latex]\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\nEvaluate [latex]\\mathrm{ln}\\left(-500\\right)[\/latex].\r\n\r\n[reveal-answer q=\"342695\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"342695\"]It is not possible to take the logarithm of a negative number in the set of real numbers.[\/hidden-answer]\r\n\r\n[ohm_question]35022[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Evaluate logarithms with and without a calculator.<\/li>\n<li>Evaluate logarithms with base 10 and base e.<\/li>\n<\/ul>\n<\/div>\n<p>Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider [latex]{\\mathrm{log}}_{2}8[\/latex]. We ask, &#8220;To what exponent must 2\u00a0be raised in order to get 8?&#8221; Because we already know [latex]{2}^{3}=8[\/latex], it follows that [latex]{\\mathrm{log}}_{2}8=3[\/latex].<\/p>\n<p>Now consider solving [latex]{\\mathrm{log}}_{7}49[\/latex] and [latex]{\\mathrm{log}}_{3}27[\/latex] mentally.<\/p>\n<ul>\n<li>We ask, &#8220;To what exponent must 7 be raised in order to get 49?&#8221; We know [latex]{7}^{2}=49[\/latex]. Therefore, [latex]{\\mathrm{log}}_{7}49=2[\/latex].<\/li>\n<li>We ask, &#8220;To what exponent must 3 be raised in order to get 27?&#8221; We know [latex]{3}^{3}=27[\/latex]. Therefore, [latex]{\\mathrm{log}}_{3}27=3[\/latex].<\/li>\n<\/ul>\n<p>Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let\u2019s evaluate [latex]{\\mathrm{log}}_{\\frac{2}{3}}\\frac{4}{9}[\/latex] mentally.<\/p>\n<ul>\n<li>We ask, &#8220;To what exponent must [latex]\\frac{2}{3}[\/latex] be raised in order to get [latex]\\frac{4}{9}[\/latex]? &#8221; We know [latex]{2}^{2}=4[\/latex] and [latex]{3}^{2}=9[\/latex], so [latex]{\\left(\\frac{2}{3}\\right)}^{2}=\\frac{4}{9}[\/latex]. Therefore, [latex]{\\mathrm{log}}_{\\frac{2}{3}}\\left(\\frac{4}{9}\\right)=2[\/latex].<\/li>\n<\/ul>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>It may be tempting to use your calculator to evaluate these logarithms but try to evaluate them mentally as it will aid your understanding of what a logarithm is, and will help you navigate more complicated situations.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a logarithm of the form [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], evaluate it mentally<\/h3>\n<ol>\n<li>Rewrite the argument <em>x<\/em>\u00a0as a power of <em>b<\/em>: [latex]{b}^{y}=x[\/latex].<\/li>\n<li>Use previous knowledge of powers of <em>b<\/em>\u00a0to identify <em>y<\/em>\u00a0by asking, &#8220;To what exponent should <em>b<\/em>\u00a0be raised in order to get <em>x<\/em>?&#8221;<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving Logarithms Mentally<\/h3>\n<p>Solve [latex]y={\\mathrm{log}}_{4}\\left(64\\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=\"q879580\">Show Solution<\/span><\/p>\n<div id=\"q879580\" class=\"hidden-answer\" style=\"display: none\">\n<p>First we rewrite the logarithm in exponential form: [latex]{4}^{y}=64[\/latex]. Next, we ask, &#8220;To what exponent must 4 be raised in order to get 64?&#8221;<\/p>\n<p>We know [latex]{4}^{3}=64[\/latex]<\/p>\n<p>Therefore,<\/p>\n<p>[latex]\\mathrm{log}{}_{4}\\left(64\\right)=3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve [latex]y={\\mathrm{log}}_{121}\\left(11\\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=\"q143125\">Show Solution<\/span><\/p>\n<div id=\"q143125\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{\\mathrm{log}}_{121}\\left(11\\right)=\\frac{1}{2}[\/latex] (recall that [latex]\\sqrt{121}={\\left(121\\right)}^{\\frac{1}{2}}=11[\/latex] )<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating the Logarithm of a Reciprocal<\/h3>\n<p>Evaluate [latex]y={\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\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=\"q861965\">Show Solution<\/span><\/p>\n<div id=\"q861965\" class=\"hidden-answer\" style=\"display: none\">\n<p>First we rewrite the logarithm in exponential form: [latex]{3}^{y}=\\frac{1}{27}[\/latex]. Next, we ask, &#8220;To what exponent must 3 be raised in order to get [latex]\\frac{1}{27}[\/latex]&#8220;?<\/p>\n<p>We know [latex]{3}^{3}=27[\/latex], but what must we do to get the reciprocal, [latex]\\frac{1}{27}[\/latex]? Recall from working with exponents that [latex]{b}^{-a}=\\frac{1}{{b}^{a}}[\/latex]. We use this information to write<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{3}^{-3}=\\frac{1}{{3}^{3}}=\\frac{1}{27}\\hfill \\end{array}[\/latex]<\/p>\n<p>Therefore, [latex]{\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)=-3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Evaluate [latex]y={\\mathrm{log}}_{2}\\left(\\frac{1}{32}\\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=\"q765423\">Show Solution<\/span><\/p>\n<div id=\"q765423\" class=\"hidden-answer\" style=\"display: none\">[latex]{\\mathrm{log}}_{2}\\left(\\frac{1}{32}\\right)=-5[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm15905\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15905&theme=oea&iframe_resize_id=ohm15905&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Using Common Logarithms<\/h2>\n<p>Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expressions [latex]\\text{log}(x)[\/latex] means [latex]\\text{log}_{10}(x).[\/latex] We call a base-[latex]10[\/latex] logarithm a\u00a0<strong>common logarithm.\u00a0<\/strong>Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Definition of the Common Logarithm<\/h3>\n<p>A <strong>common logarithm<\/strong> is a logarithm with base [latex]10[\/latex]. We write [latex]\\text{log}_{10}(x).[\/latex] simply as [latex]\\text{log}(x)[\/latex]. The common logarithm of a positive number [latex]x[\/latex]\u00a0satisfies the following definition:<\/p>\n<p>For [latex]x>0[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]y=\\text{log}(x)\\text{ is equivalent to }{10}^{y}=x[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left;\">We read [latex]\\text{log}(x)[\/latex] as, &#8220;the logarithm with base [latex]10[\/latex]\u00a0of [latex]x[\/latex]&#8221; or &#8220;the common logarithm of [latex]x[\/latex].&#8221;<\/p>\n<p>The logarithm [latex]y[\/latex] is the exponent to which [latex]10[\/latex] must be raised to get [latex]x[\/latex].<\/p>\n<p>Since the functions [latex]y=10^{x}[\/latex] and [latex]y=\\mathrm{log}\\left(x\\right)[\/latex] are inverse functions, [latex]\\mathrm{log}\\left({10}^{x}\\right)=x[\/latex] for all [latex]x[\/latex]\u00a0and [latex]10^{\\mathrm{log}\\left(x\\right)}=x[\/latex] for [latex]x>0[\/latex].<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a common logarithm Of the form [latex]y=\\mathrm{log}\\left(x\\right)[\/latex], evaluate it using a calculator+<\/h3>\n<ol>\n<li>Press <strong>[LOG]<\/strong>.<\/li>\n<li>Enter the value given for <em>x<\/em>, followed by <strong>[ ) ]<\/strong>.<\/li>\n<li>Press <strong>[ENTER]<\/strong>.<\/li>\n<\/ol>\n<\/div>\n<h2>Using Natural Logarithms<\/h2>\n<p>The most frequently used base for logarithms is <em>e<\/em>. Base <em>e<\/em>\u00a0logarithms are important in calculus and some scientific applications; they are called <strong>natural logarithms<\/strong>. The base <em>e<\/em>\u00a0logarithm, [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex], has its own notation, [latex]\\mathrm{ln}\\left(x\\right)[\/latex].<\/p>\n<p>Most values of [latex]\\mathrm{ln}\\left(x\\right)[\/latex] can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, [latex]\\mathrm{ln}1=0[\/latex]. For other natural logarithms, we can use the [latex]\\mathrm{ln}[\/latex] key that can be found on most scientific calculators. We can also find the natural logarithm of any power of <em>e<\/em>\u00a0using the inverse property of logarithms.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Definition of the Natural Logarithm<\/h3>\n<p>A <strong>natural logarithm<\/strong> is a logarithm with base <em>e<\/em>. We write [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex] simply as [latex]\\mathrm{ln}\\left(x\\right)[\/latex]. The natural logarithm of a positive number <em>x<\/em>\u00a0satisfies the following definition:<\/p>\n<p>For [latex]x>0[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]y=\\mathrm{ln}\\left(x\\right)\\text{ is equivalent to }{e}^{y}=x[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>We read [latex]\\mathrm{ln}\\left(x\\right)[\/latex] as, &#8220;the logarithm with base <em>e<\/em>\u00a0of <em>x<\/em>&#8221; or &#8220;the natural logarithm of <em>x<\/em>.&#8221;<\/p>\n<p>The logarithm [latex]y[\/latex]\u00a0is the exponent to which <em>e<\/em>\u00a0must be raised to get [latex]x[\/latex].<\/p>\n<p>Since the functions [latex]y=e^{x}[\/latex] and [latex]y=\\mathrm{ln}\\left(x\\right)[\/latex] are inverse functions, [latex]\\mathrm{ln}\\left({e}^{x}\\right)=x[\/latex] for all <em>x<\/em>\u00a0and [latex]e^{\\mathrm{ln}\\left(x\\right)}=x[\/latex] for [latex]x>0[\/latex].<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a natural logarithm Of the form [latex]y=\\mathrm{ln}\\left(x\\right)[\/latex], evaluate it using a calculator+<\/h3>\n<ol>\n<li>Press <strong>[LN]<\/strong>.<\/li>\n<li>Enter the value given for <em>x<\/em>, followed by <strong>[ ) ]<\/strong>.<\/li>\n<li>Press <strong>[ENTER]<\/strong>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating a Natural Logarithm Using a Calculator<\/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=\"q810207\">Show Solution<\/span><\/p>\n<div id=\"q810207\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>Press <strong>[LN]<\/strong>.<\/li>\n<li>Enter 500, followed by <strong>[ ) ]<\/strong>.<\/li>\n<li>Press <strong>[ENTER]<\/strong>.<\/li>\n<\/ul>\n<p>Rounding to four decimal places, [latex]\\mathrm{ln}\\left(500\\right)\\approx 6.2146[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Evaluate [latex]\\mathrm{ln}\\left(-500\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q342695\">Show Solution<\/span><\/p>\n<div id=\"q342695\" class=\"hidden-answer\" style=\"display: none\">It is not possible to take the logarithm of a negative number in the set of real numbers.<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm35022\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=35022&theme=oea&iframe_resize_id=ohm35022&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-268\">\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>Question ID 35022. <strong>Authored by<\/strong>: Smart,Jim, mb Sousa,James. <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>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><\/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":395986,"menu_order":30,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 35022\",\"author\":\"Smart,Jim, mb Sousa,James\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen 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