{"id":3898,"date":"2021-05-20T18:29:56","date_gmt":"2021-05-20T18:29:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/latexreview-for-lh-o_hat-pitals-rule-latex\/"},"modified":"2021-07-03T19:08:38","modified_gmt":"2021-07-03T19:08:38","slug":"latexreview-for-lh-o_hat-pitals-rule-latex","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/latexreview-for-lh-o_hat-pitals-rule-latex\/","title":{"raw":"Skills Review for L'H\u00f4pital's Rule","rendered":"Skills Review for L&#8217;H\u00f4pital&#8217;s Rule"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Convert from logarithmic form to exponential form&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:{&quot;1&quot;:2,&quot;2&quot;:13624051},&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Convert from logarithmic form to exponential form<\/span><\/li>\r\n \t<li>Define and use the power rule for logarithms to rewrite expressions<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the L'Hopital's Rule section, some of the concepts require working with natural logarithms. Here we will review how to convert from logarithmic to exponential form in addition to the power property for logarithms.\r\n<h2>Convert from Logarithmic to Exponential Form<\/h2>\r\nWe can express the relationship between logarithmic form and its corresponding exponential form as follows:\r\n\r\n[latex]{\\mathrm{log}}_{b}\\left(x\\right)=y\\Leftrightarrow {b}^{y}=x,\\text{}b&gt;0,b\\ne 1[\/latex]\r\n\r\nNote that the base <em>b<\/em>\u00a0is always positive.\r\n\r\n<img class=\"aligncenter size-full wp-image-3090\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16195822\/CNX_Precalc_Figure_04_03_0042.jpg\" alt=\"Think b to the y equals x.\" width=\"487\" height=\"83\" \/>\r\n\r\nBecause a logarithm is a function, it is 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>.\r\n\r\nWe can illustrate the notation of logarithms as follows:\r\n\r\n<img class=\"aligncenter size-full wp-image-3092\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16200035\/CNX_Precalc_Figure_04_03_0032.jpg\" alt=\"logb (c) = a means b to the A power equals C.\" width=\"487\" height=\"101\" \/>\r\n\r\nNote: Sometimes, you will see the natural logarithm [latex]ln[\/latex] in place of [latex]log[\/latex]. What is important to know is that the natural logarithm\u00a0([latex]ln[\/latex]) is understood to have the number [latex]e[\/latex] as its base. The value of [latex]e[\/latex] is about 2.718.\r\n<div class=\"textbox\">\r\n<h3>How To: Given an equation in logarithmic form [latex]{\\mathrm{log}}_{b}\\left(x\\right)=y[\/latex], convert it to exponential form<\/h3>\r\n<ol>\r\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>\r\n \t<li>Rewrite [latex]{\\mathrm{log}}_{b}x=y[\/latex] as [latex]{b}^{y}=x[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Converting from Logarithmic Form to Exponential Form<\/h3>\r\nWrite the following logarithmic equations in exponential form.\r\n<ol>\r\n \t<li>[latex]{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]{\\mathrm{ln}}\\left(9\\right)=a[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"642511\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"642511\"]\r\n\r\nFirst, identify the values of <em>b<\/em>,\u00a0<em>y<\/em>, and\u00a0<em>x<\/em>. Then, write the equation in the form [latex]{b}^{y}=x[\/latex].\r\n<ol>\r\n \t<li>[latex]{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}[\/latex] 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] is equal to [latex]{6}^{\\frac{1}{2}}=\\sqrt{6}[\/latex].<\/li>\r\n \t<li>[latex]{\\mathrm{ln}}\\left(9\\right)=a[\/latex] Here, [latex]b=e[\/latex], [latex]y=a[\/latex], and [latex]x=9[\/latex]. Therefore, the equation [latex]{\\mathrm{ln}}\\left(9\\right)=a[\/latex] is equal to [latex]{e}^{a}=9[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite the following logarithmic equations in exponential form.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]{\\mathrm{log}}_{10}\\left(1,000,000\\right)=6[\/latex]<\/li>\r\n \t<li>[latex]{\\mathrm{log}}_{5}\\left(25\\right)=2[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"200815\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"200815\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]{\\mathrm{log}}_{10}\\left(1,000,000\\right)=6[\/latex] is equal to [latex]{10}^{6}=1,000,000[\/latex]<\/li>\r\n \t<li>[latex]{\\mathrm{log}}_{5}\\left(25\\right)=2[\/latex] is equal to [latex]{5}^{2}=25[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]29661[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Use the Power Property for Logarithms<\/h2>\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Power Rule for Logarithms<\/h3>\r\nThe <strong>power rule for logarithms<\/strong> can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.\r\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left({M}^{n}\\right)=n{\\mathrm{log}}_{b}M[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Expanding a Logarithm with Powers<\/h3>\r\nRewrite [latex]{\\mathrm{log}}_{2}{x}^{5}[\/latex].\r\n\r\n[reveal-answer q=\"979765\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"979765\"]\r\n\r\nThe argument is already written as a power, so we identify the exponent, 5, and the base, <em>x<\/em>, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.\r\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{2}\\left({x}^{5}\\right)=5{\\mathrm{log}}_{2}x[\/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\nRewrite [latex]\\mathrm{ln}{x}^{2}[\/latex].\r\n\r\n[reveal-answer q=\"383972\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"383972\"]\r\n\r\n[latex]2\\mathrm{ln}x[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Rewriting an Expression as a Power before Using the Power Rule<\/h3>\r\nRewrite [latex]{\\mathrm{log}}_{3}\\left(25\\right)[\/latex] using the power rule for logs.\r\n\r\n[reveal-answer q=\"984289\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"984289\"]\r\n\r\nExpressing the argument as a power, we get [latex]{\\mathrm{log}}_{3}\\left(25\\right)={\\mathrm{log}}_{3}\\left({5}^{2}\\right)[\/latex].\r\n\r\nNext we identify the exponent, 2, and the base, 5, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.\r\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{3}\\left({5}^{2}\\right)=2{\\mathrm{log}}_{3}\\left(5\\right)[\/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\nRewrite [latex]\\mathrm{ln}\\left(\\frac{1}{{x}^{2}}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"947582\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"947582\"]\r\n\r\n[latex]-2\\mathrm{ln}\\left(x\\right)[\/latex][\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Convert from logarithmic form to exponential form&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:{&quot;1&quot;:2,&quot;2&quot;:13624051},&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Convert from logarithmic form to exponential form<\/span><\/li>\n<li>Define and use the power rule for logarithms to rewrite expressions<\/li>\n<\/ul>\n<\/div>\n<p>In the L&#8217;Hopital&#8217;s Rule section, some of the concepts require working with natural logarithms. Here we will review how to convert from logarithmic to exponential form in addition to the power property for logarithms.<\/p>\n<h2>Convert from Logarithmic to Exponential Form<\/h2>\n<p>We can express the relationship between logarithmic form and its corresponding exponential form as follows:<\/p>\n<p>[latex]{\\mathrm{log}}_{b}\\left(x\\right)=y\\Leftrightarrow {b}^{y}=x,\\text{}b>0,b\\ne 1[\/latex]<\/p>\n<p>Note that the base <em>b<\/em>\u00a0is always positive.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-3090\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16195822\/CNX_Precalc_Figure_04_03_0042.jpg\" alt=\"Think b to the y equals x.\" width=\"487\" height=\"83\" \/><\/p>\n<p>Because a logarithm is a function, it is 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>We can illustrate the notation of logarithms as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-3092\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/16200035\/CNX_Precalc_Figure_04_03_0032.jpg\" alt=\"logb (c) = a means b to the A power equals C.\" width=\"487\" height=\"101\" \/><\/p>\n<p>Note: Sometimes, you will see the natural logarithm [latex]ln[\/latex] in place of [latex]log[\/latex]. What is important to know is that the natural logarithm\u00a0([latex]ln[\/latex]) is understood to have the number [latex]e[\/latex] as its base. The value of [latex]e[\/latex] is about 2.718.<\/p>\n<div class=\"textbox\">\n<h3>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>\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 exercises\">\n<h3>Example: Converting from Logarithmic Form to Exponential Form<\/h3>\n<p>Write the following logarithmic equations in exponential form.<\/p>\n<ol>\n<li>[latex]{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}[\/latex]<\/li>\n<li>[latex]{\\mathrm{ln}}\\left(9\\right)=a[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q642511\">Show Solution<\/span><\/p>\n<div id=\"q642511\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, identify the values of <em>b<\/em>,\u00a0<em>y<\/em>, and\u00a0<em>x<\/em>. Then, write the equation in the form [latex]{b}^{y}=x[\/latex].<\/p>\n<ol>\n<li>[latex]{\\mathrm{log}}_{6}\\left(\\sqrt{6}\\right)=\\frac{1}{2}[\/latex] 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] is equal to [latex]{6}^{\\frac{1}{2}}=\\sqrt{6}[\/latex].<\/li>\n<li>[latex]{\\mathrm{ln}}\\left(9\\right)=a[\/latex] Here, [latex]b=e[\/latex], [latex]y=a[\/latex], and [latex]x=9[\/latex]. Therefore, the equation [latex]{\\mathrm{ln}}\\left(9\\right)=a[\/latex] is equal to [latex]{e}^{a}=9[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write the following logarithmic equations in exponential form.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]{\\mathrm{log}}_{10}\\left(1,000,000\\right)=6[\/latex]<\/li>\n<li>[latex]{\\mathrm{log}}_{5}\\left(25\\right)=2[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q200815\">Show Solution<\/span><\/p>\n<div id=\"q200815\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]{\\mathrm{log}}_{10}\\left(1,000,000\\right)=6[\/latex] is equal to [latex]{10}^{6}=1,000,000[\/latex]<\/li>\n<li>[latex]{\\mathrm{log}}_{5}\\left(25\\right)=2[\/latex] is equal to [latex]{5}^{2}=25[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm29661\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29661&theme=oea&iframe_resize_id=ohm29661&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Use the Power Property for Logarithms<\/h2>\n<div class=\"textbox\">\n<h3>A General Note: The Power Rule for Logarithms<\/h3>\n<p>The <strong>power rule for logarithms<\/strong> can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.<\/p>\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left({M}^{n}\\right)=n{\\mathrm{log}}_{b}M[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Expanding a Logarithm with Powers<\/h3>\n<p>Rewrite [latex]{\\mathrm{log}}_{2}{x}^{5}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q979765\">Show Solution<\/span><\/p>\n<div id=\"q979765\" class=\"hidden-answer\" style=\"display: none\">\n<p>The argument is already written as a power, so we identify the exponent, 5, and the base, <em>x<\/em>, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.<\/p>\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{2}\\left({x}^{5}\\right)=5{\\mathrm{log}}_{2}x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Rewrite [latex]\\mathrm{ln}{x}^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q383972\">Show Solution<\/span><\/p>\n<div id=\"q383972\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]2\\mathrm{ln}x[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Rewriting an Expression as a Power before Using the Power Rule<\/h3>\n<p>Rewrite [latex]{\\mathrm{log}}_{3}\\left(25\\right)[\/latex] using the power rule for logs.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q984289\">Show Solution<\/span><\/p>\n<div id=\"q984289\" class=\"hidden-answer\" style=\"display: none\">\n<p>Expressing the argument as a power, we get [latex]{\\mathrm{log}}_{3}\\left(25\\right)={\\mathrm{log}}_{3}\\left({5}^{2}\\right)[\/latex].<\/p>\n<p>Next we identify the exponent, 2, and the base, 5, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.<\/p>\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{3}\\left({5}^{2}\\right)=2{\\mathrm{log}}_{3}\\left(5\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Rewrite [latex]\\mathrm{ln}\\left(\\frac{1}{{x}^{2}}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q947582\">Show Solution<\/span><\/p>\n<div id=\"q947582\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-2\\mathrm{ln}\\left(x\\right)[\/latex]<\/p><\/div>\n<\/div>\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-3898\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Modification and Revision . <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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