{"id":3890,"date":"2021-05-20T18:29:55","date_gmt":"2021-05-20T18:29:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-derivatives-of-exponential-and-logarithmic-equations\/"},"modified":"2021-07-03T18:58:25","modified_gmt":"2021-07-03T18:58:25","slug":"review-for-derivatives-of-exponential-and-logarithmic-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-derivatives-of-exponential-and-logarithmic-equations\/","title":{"raw":"Skills Review for Derivatives of Exponential and Logarithmic Functions","rendered":"Skills Review for Derivatives of Exponential and Logarithmic Functions"},"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;Factor out the greatest common factor of a polynomial&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,16573901],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Factor out the common exponential or logarithmic factors of a polynomial<\/span><\/li>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Combine the product, power, and quotient rules to expand logarithmic expressions&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;:14281427},&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Combine the product, power, and quotient rules to expand logarithmic expressions<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the\u00a0<em>Derivatives of Exponential and Logarithmic Equations<\/em>\u00a0section, you will learn how to take derivatives of exponential and logarithmic functions. Here we will review how to factor expressions that contain common exponential and logarithmic components in their terms. To prepare you for logarithmic differentiation, we will review the properties of logarithms and use them to expand logarithmic expressions.\r\n<h2>Factoring Exponential and Logarithmic Expressions<\/h2>\r\nThe greatest common factor of an algebraic expression can sometimes be an exponential or logarithmic quantity. When factoring expressions that contain common exponential or logarithmic quantities in each of their terms, these common quantities can be factored from the expression.\r\n\r\nIn the expression [latex]x^2e^{3x}+7e^{3x}[\/latex], since [latex]e^{3x}[\/latex] is present in both terms, it can be factored from the expression; the factored result is\u00a0[latex]e^{3x}(x^2+7)[\/latex].\r\n\r\nIn the expression [latex]3x\\ln(5)2^{x}+2\\ln(5)2^{x}[\/latex], since\u00a0[latex]\\ln(5)[\/latex] and [latex]2^{x}[\/latex] are present in both terms, they can be factored from the expression; the factored result is\u00a0[latex]\\ln(5)2^{x}(3x+2)[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example: FactorIng An Exponential Expression<\/h3>\r\nFactor [latex]6({5}^{x})+45{x}^{2}({5}^{x})+21x({5}^{x})[\/latex].\r\n\r\n[reveal-answer q=\"113189\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"113189\"]\r\n\r\nAll three terms contain [latex]5^x[\/latex].\r\n\r\nNext, factor\u00a0[latex]5^x[\/latex] from each term.\r\n\r\nAfter factoring, the result is\u00a0[latex]{5}^{x}(6+45{x}^{2}+21x)[\/latex].\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Factoring an Exponential and Logarithmic Expression<\/h3>\r\nFactor [latex]-6\\ln(7)({7}^{x})+2{x}^{2}\\ln(7)({7}^{x})[\/latex].\r\n\r\n[reveal-answer q=\"113190\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"113190\"]\r\n\r\nBoth terms contain [latex]7^x[\/latex] and\u00a0[latex]\\ln(7)[\/latex].\r\n\r\nNext, factor\u00a0[latex]7^x[\/latex] and [latex]\\ln(7)[\/latex] from each term.\r\n\r\nAfter factoring, the result is\u00a0[latex]\\ln(7){7}^{x}(-6+2{x}^{2})[\/latex].\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0FactorIng A Rational Exponential Expression<\/h3>\r\nFactor [latex]\\dfrac{3x^2({8}^{x})+10x({8}^{x})}{8^{x}}[\/latex].\r\n\r\n[reveal-answer q=\"113191\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"113191\"]\r\n\r\nBoth terms in the numerator contain [latex]8^x[\/latex] and\u00a0[latex]x[\/latex] (we can factor out\u00a0[latex]x[\/latex] to the lowest power present among the given terms in the numerator).\r\n\r\nNext, factor\u00a0[latex]8^x[\/latex] and\u00a0[latex]x[\/latex] from each term in the numerator.\r\n\r\nAfter factoring, the result is [latex]\\frac{x8^x(3x+10)}{8^{x}}[\/latex].\r\n\r\nNotice the\u00a0[latex]8^x[\/latex] in the numerator and\u00a0[latex]8^x[\/latex] in the denominator cancel.\r\n\r\nThe final answer is\u00a0[latex]x(3x+10)[\/latex].\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<h2>Use Properties of Logarithms to Expand Logarithmic Expressions<\/h2>\r\nThere are a variety of logarithmic properties that allow us to expand a logarithmic expression.\r\n<div class=\"textbox\">\r\n<h3>The Product Rule for Logarithms<\/h3>\r\nThe <strong>product rule for logarithms<\/strong> can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.\r\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left(MN\\right)={\\mathrm{log}}_{b}\\left(M\\right)+{\\mathrm{log}}_{b}\\left(N\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137733855\" class=\"note textbox\">\r\n<h3 class=\"title\">\u00a0The Quotient Rule for Logarithms<\/h3>\r\nThe <strong>quotient rule for logarithms<\/strong> can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.\r\n<div style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left(\\dfrac{M}{N}\\right)={\\mathrm{log}}_{b}M-{\\mathrm{log}}_{b}N[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>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\nTaken together, the product rule, quotient rule, and power rule are often called \"properties of logs.\" Sometimes we apply more than one rule in order to expand an expression. For example:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{\\mathrm{log}}_{b}\\left(\\frac{6x}{y}\\right)\\hfill &amp; ={\\mathrm{log}}_{b}\\left(6x\\right)-{\\mathrm{log}}_{b}y\\hfill \\\\ \\hfill &amp; ={\\mathrm{log}}_{b}6+{\\mathrm{log}}_{b}x-{\\mathrm{log}}_{b}y\\hfill \\end{array}[\/latex]<\/p>\r\nRemember, however, that we can only expand logarithms with products, quotients, powers, and roots\u2014never with addition or subtraction inside the argument of the logarithm.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Expanding An Expression Using Properties of Logs<\/h3>\r\nRewrite [latex]\\mathrm{ln}\\left(\\frac{{x}^{4}y}{7}\\right)[\/latex] as a sum or difference of logs.\r\n\r\n[reveal-answer q=\"526416\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"526416\"]\r\n\r\nFirst, because we have a quotient of two expressions, we can use the quotient rule:\r\n\r\n[latex]\\mathrm{ln}\\left(\\frac{{x}^{4}y}{7}\\right)=\\mathrm{ln}\\left({x}^{4}y\\right)-\\mathrm{ln}\\left(7\\right)[\/latex]\r\n\r\nThen seeing the product in the first term, we use the product rule:\r\n\r\n[latex]\\mathrm{ln}\\left({x}^{4}y\\right)-\\mathrm{ln}\\left(7\\right)=\\mathrm{ln}\\left({x}^{4}\\right)+\\mathrm{ln}\\left(y\\right)-\\mathrm{ln}\\left(7\\right)[\/latex]\r\n\r\nFinally, we use the power rule on the first term:\r\n\r\n[latex]\\mathrm{ln}\\left({x}^{4}\\right)+\\mathrm{ln}\\left(y\\right)-\\mathrm{ln}\\left(7\\right)=4\\mathrm{ln}\\left(x\\right)+\\mathrm{ln}\\left(y\\right)-\\mathrm{ln}\\left(7\\right)[\/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\nExpand [latex]\\mathrm{log}\\left(\\frac{{x}^{2}{y}^{3}}{{z}^{4}}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"722800\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"722800\"]\r\n\r\n[latex]2\\mathrm{log}x+3\\mathrm{log}y - 4\\mathrm{log}z[\/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\n[ohm_question]35034[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Expanding An Expression Using Properties of Logs<\/h3>\r\nExpand [latex]\\mathrm{log}\\left(x^3\\sqrt{y}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"914877\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"914877\"]\r\n\r\n[latex]\\begin{array}{l}\\mathrm{log}\\left(x^3\\sqrt{y}\\right)\\hfill &amp; =\\mathrm{log}(x^3{y}^{\\frac{1}{2}})\\hfill \\\\ \\hfill &amp; =\\mathrm{log}x^3 + \\mathrm{log}{y}^{\\frac{1}{2}}\\hfill \\\\ \\hfill &amp; =\\mathrm3{log}x+\\frac{1}{2}\\mathrm{log}y\\hfill\\end{array}[\/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\nExpand [latex]\\mathrm{ln}\\left(x^{4}\\sqrt[3]{{y}^{2}}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"2296\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"2296\"]\r\n\r\n[latex]4\\mathrm{ln}x+\\frac{2}{3}\\mathrm{ln}y[\/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;Factor out the greatest common factor of a polynomial&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,16573901],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Factor out the common exponential or logarithmic factors of a polynomial<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Combine the product, power, and quotient rules to expand logarithmic expressions&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;:14281427},&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Combine the product, power, and quotient rules to expand logarithmic expressions<\/span><\/li>\n<\/ul>\n<\/div>\n<p>In the\u00a0<em>Derivatives of Exponential and Logarithmic Equations<\/em>\u00a0section, you will learn how to take derivatives of exponential and logarithmic functions. Here we will review how to factor expressions that contain common exponential and logarithmic components in their terms. To prepare you for logarithmic differentiation, we will review the properties of logarithms and use them to expand logarithmic expressions.<\/p>\n<h2>Factoring Exponential and Logarithmic Expressions<\/h2>\n<p>The greatest common factor of an algebraic expression can sometimes be an exponential or logarithmic quantity. When factoring expressions that contain common exponential or logarithmic quantities in each of their terms, these common quantities can be factored from the expression.<\/p>\n<p>In the expression [latex]x^2e^{3x}+7e^{3x}[\/latex], since [latex]e^{3x}[\/latex] is present in both terms, it can be factored from the expression; the factored result is\u00a0[latex]e^{3x}(x^2+7)[\/latex].<\/p>\n<p>In the expression [latex]3x\\ln(5)2^{x}+2\\ln(5)2^{x}[\/latex], since\u00a0[latex]\\ln(5)[\/latex] and [latex]2^{x}[\/latex] are present in both terms, they can be factored from the expression; the factored result is\u00a0[latex]\\ln(5)2^{x}(3x+2)[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: FactorIng An Exponential Expression<\/h3>\n<p>Factor [latex]6({5}^{x})+45{x}^{2}({5}^{x})+21x({5}^{x})[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q113189\">Show Solution<\/span><\/p>\n<div id=\"q113189\" class=\"hidden-answer\" style=\"display: none\">\n<p>All three terms contain [latex]5^x[\/latex].<\/p>\n<p>Next, factor\u00a0[latex]5^x[\/latex] from each term.<\/p>\n<p>After factoring, the result is\u00a0[latex]{5}^{x}(6+45{x}^{2}+21x)[\/latex].<\/p>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Factoring an Exponential and Logarithmic Expression<\/h3>\n<p>Factor [latex]-6\\ln(7)({7}^{x})+2{x}^{2}\\ln(7)({7}^{x})[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q113190\">Show Solution<\/span><\/p>\n<div id=\"q113190\" class=\"hidden-answer\" style=\"display: none\">\n<p>Both terms contain [latex]7^x[\/latex] and\u00a0[latex]\\ln(7)[\/latex].<\/p>\n<p>Next, factor\u00a0[latex]7^x[\/latex] and [latex]\\ln(7)[\/latex] from each term.<\/p>\n<p>After factoring, the result is\u00a0[latex]\\ln(7){7}^{x}(-6+2{x}^{2})[\/latex].<\/p>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0FactorIng A Rational Exponential Expression<\/h3>\n<p>Factor [latex]\\dfrac{3x^2({8}^{x})+10x({8}^{x})}{8^{x}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q113191\">Show Solution<\/span><\/p>\n<div id=\"q113191\" class=\"hidden-answer\" style=\"display: none\">\n<p>Both terms in the numerator contain [latex]8^x[\/latex] and\u00a0[latex]x[\/latex] (we can factor out\u00a0[latex]x[\/latex] to the lowest power present among the given terms in the numerator).<\/p>\n<p>Next, factor\u00a0[latex]8^x[\/latex] and\u00a0[latex]x[\/latex] from each term in the numerator.<\/p>\n<p>After factoring, the result is [latex]\\frac{x8^x(3x+10)}{8^{x}}[\/latex].<\/p>\n<p>Notice the\u00a0[latex]8^x[\/latex] in the numerator and\u00a0[latex]8^x[\/latex] in the denominator cancel.<\/p>\n<p>The final answer is\u00a0[latex]x(3x+10)[\/latex].<\/p>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Use Properties of Logarithms to Expand Logarithmic Expressions<\/h2>\n<p>There are a variety of logarithmic properties that allow us to expand a logarithmic expression.<\/p>\n<div class=\"textbox\">\n<h3>The Product Rule for Logarithms<\/h3>\n<p>The <strong>product rule for logarithms<\/strong> can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.<\/p>\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left(MN\\right)={\\mathrm{log}}_{b}\\left(M\\right)+{\\mathrm{log}}_{b}\\left(N\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137733855\" class=\"note textbox\">\n<h3 class=\"title\">\u00a0The Quotient Rule for Logarithms<\/h3>\n<p>The <strong>quotient rule for logarithms<\/strong> can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.<\/p>\n<div style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left(\\dfrac{M}{N}\\right)={\\mathrm{log}}_{b}M-{\\mathrm{log}}_{b}N[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>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<p>Taken together, the product rule, quotient rule, and power rule are often called &#8220;properties of logs.&#8221; Sometimes we apply more than one rule in order to expand an expression. For example:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{\\mathrm{log}}_{b}\\left(\\frac{6x}{y}\\right)\\hfill & ={\\mathrm{log}}_{b}\\left(6x\\right)-{\\mathrm{log}}_{b}y\\hfill \\\\ \\hfill & ={\\mathrm{log}}_{b}6+{\\mathrm{log}}_{b}x-{\\mathrm{log}}_{b}y\\hfill \\end{array}[\/latex]<\/p>\n<p>Remember, however, that we can only expand logarithms with products, quotients, powers, and roots\u2014never with addition or subtraction inside the argument of the logarithm.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Expanding An Expression Using Properties of Logs<\/h3>\n<p>Rewrite [latex]\\mathrm{ln}\\left(\\frac{{x}^{4}y}{7}\\right)[\/latex] as a sum or difference of logs.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q526416\">Show Solution<\/span><\/p>\n<div id=\"q526416\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, because we have a quotient of two expressions, we can use the quotient rule:<\/p>\n<p>[latex]\\mathrm{ln}\\left(\\frac{{x}^{4}y}{7}\\right)=\\mathrm{ln}\\left({x}^{4}y\\right)-\\mathrm{ln}\\left(7\\right)[\/latex]<\/p>\n<p>Then seeing the product in the first term, we use the product rule:<\/p>\n<p>[latex]\\mathrm{ln}\\left({x}^{4}y\\right)-\\mathrm{ln}\\left(7\\right)=\\mathrm{ln}\\left({x}^{4}\\right)+\\mathrm{ln}\\left(y\\right)-\\mathrm{ln}\\left(7\\right)[\/latex]<\/p>\n<p>Finally, we use the power rule on the first term:<\/p>\n<p>[latex]\\mathrm{ln}\\left({x}^{4}\\right)+\\mathrm{ln}\\left(y\\right)-\\mathrm{ln}\\left(7\\right)=4\\mathrm{ln}\\left(x\\right)+\\mathrm{ln}\\left(y\\right)-\\mathrm{ln}\\left(7\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Expand [latex]\\mathrm{log}\\left(\\frac{{x}^{2}{y}^{3}}{{z}^{4}}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q722800\">Show Solution<\/span><\/p>\n<div id=\"q722800\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]2\\mathrm{log}x+3\\mathrm{log}y - 4\\mathrm{log}z[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm35034\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=35034&theme=oea&iframe_resize_id=ohm35034&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Expanding An Expression Using Properties of Logs<\/h3>\n<p>Expand [latex]\\mathrm{log}\\left(x^3\\sqrt{y}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q914877\">Show Solution<\/span><\/p>\n<div id=\"q914877\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{array}{l}\\mathrm{log}\\left(x^3\\sqrt{y}\\right)\\hfill & =\\mathrm{log}(x^3{y}^{\\frac{1}{2}})\\hfill \\\\ \\hfill & =\\mathrm{log}x^3 + \\mathrm{log}{y}^{\\frac{1}{2}}\\hfill \\\\ \\hfill & =\\mathrm3{log}x+\\frac{1}{2}\\mathrm{log}y\\hfill\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Expand [latex]\\mathrm{ln}\\left(x^{4}\\sqrt[3]{{y}^{2}}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q2296\">Show Solution<\/span><\/p>\n<div id=\"q2296\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]4\\mathrm{ln}x+\\frac{2}{3}\\mathrm{ln}y[\/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-3890\">\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>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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><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 <\/section>","protected":false},"author":17533,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Modification and Revision\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra Corequisite\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/precalculus\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3890","chapter","type-chapter","status-publish","hentry"],"part":3089,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3890","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3890\/revisions"}],"predecessor-version":[{"id":4066,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3890\/revisions\/4066"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/3089"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3890\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3890"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3890"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3890"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3890"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}