{"id":481,"date":"2025-02-13T19:45:21","date_gmt":"2025-02-13T19:45:21","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integration-of-exponential-logarithmic-and-hyperbolic-functions-background-youll-need-1\/"},"modified":"2025-02-13T19:45:21","modified_gmt":"2025-02-13T19:45:21","slug":"integration-of-exponential-logarithmic-and-hyperbolic-functions-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integration-of-exponential-logarithmic-and-hyperbolic-functions-background-youll-need-1\/","title":{"raw":"Integration of Exponential, Logarithmic, and Hyperbolic Functions: Background You'll Need 1","rendered":"Integration of Exponential, Logarithmic, and Hyperbolic Functions: Background You&#8217;ll Need 1"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Use rules for multiplying, raising to a power, and dividing to simplify logarithmic expressions<\/li>\n<\/ul>\n<\/section>\n<p>Logarithms are essential tools in calculus, particularly when dealing with integrals and exponential functions. Mastering how to condense logarithmic expressions using fundamental rules will simplify complex calculations and enhance your understanding of these concepts.<\/p>\n<h2>Simplify Logarithmic Expressions<\/h2>\n<p>Simplifying logarithmic expressions involves applying specific rules to transform intricate terms into more manageable ones. This process is crucial for effectively solving equations in calculus and other mathematical disciplines.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>rules for logarithms<\/h3>\n<ul>\n\t<li><strong>Product Rule:<\/strong> The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.<center>[latex]{\\mathrm{log}}_{b}\\left(M \\times N\\right)={\\mathrm{log}}_{b}\\left(M\\right)+{\\mathrm{log}}_{b}\\left(N\\right)\\text{ for }b&gt;0[\/latex]<\/center><\/li>\n\t<li><strong>Quotient Rule:<\/strong> The quotient rule for logarithms can be used to simplify a logarithm of a quotient by rewriting it as the difference of individual logarithms.<center>[latex]{\\mathrm{log}}_{b}\\left(\\frac{M}{N}\\right)={\\mathrm{log}}_{b}M-{\\mathrm{log}}_{b}N[\/latex]<\/center><\/li>\n\t<li><strong>Power Rule:<\/strong> The power rule for logarithms 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.<center>[latex]{\\mathrm{log}}_{b}\\left({M}^{n}\\right)=n{\\mathrm{log}}_{b}M[\/latex]<\/center><\/li>\n<\/ul>\n<\/section>\n<p>We can use the rules of logarithms to condense sums, differences, and products with the same base as a single logarithm. It is important to remember that the logarithms must have the same base to be combined.<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Given a sum, difference, or product of logarithms with the same base, write an equivalent expression as a single logarithm<\/strong><\/p>\n<ol>\n\t<li><strong>Apply the Power Rule:<\/strong> Identify and rewrite any terms that are powers of factors as the logarithm of a power.<\/li>\n\t<li><strong>Apply the Product and Quotient Rules:<\/strong> From left to right, apply the product and quotient properties to rewrite sums of logarithms as the logarithm of a product and differences as the logarithm of a quotient.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">\n<p>Use the power rule for logs to rewrite [latex]4\\mathrm{ln}\\left(x\\right)[\/latex] as a single logarithm with a leading coefficient of 1.<\/p>\n<p>[reveal-answer q=\"959478\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"959478\"]<\/p>\n<p>Because the logarithm of a power is the product of the exponent times the logarithm of the base, it follows that the product of a number and a logarithm can be written as a power. <br>\n<br>\nFor the expression [latex]4\\mathrm{ln}\\left(x\\right)[\/latex], we identify the factor, [latex]4[\/latex], as the exponent and the argument, [latex]x[\/latex], as the base and rewrite the product as a logarithm of a power:<\/p>\n<p style=\"text-align: center;\">[latex]4\\mathrm{ln}\\left(x\\right)=\\mathrm{ln}\\left({x}^{4}\\right)[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Use the power rule for logs to rewrite [latex]2{\\mathrm{log}}_{3}4[\/latex] as a single logarithm with a leading coefficient of 1.<\/p>\n<p>[reveal-answer q=\"720709\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"720709\"]<\/p>\n<p>[latex]{\\mathrm{log}}_{3}16[\/latex][\/hidden-answer]<\/p>\n<\/section>\n<p>In our next few examples, we will use a combination of logarithm rules to condense logarithms.<\/p>\n<section class=\"textbox example\">\n<p>Write [latex]{\\mathrm{log}}_{3}\\left(5\\right)+{\\mathrm{log}}_{3}\\left(8\\right)-{\\mathrm{log}}_{3}\\left(2\\right)[\/latex] as a single logarithm.<\/p>\n<p>[reveal-answer q=\"484876\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"484876\"]<\/p>\n<p>From left to right, since we have the addition of two logs, we first use the product rule:<\/p>\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{3}\\left(5\\right)+{\\mathrm{log}}_{3}\\left(8\\right)={\\mathrm{log}}_{3}\\left(5\\cdot 8\\right)={\\mathrm{log}}_{3}\\left(40\\right)[\/latex]<\/p>\n<p>This simplifies our original expression to:<\/p>\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{3}\\left(40\\right)-{\\mathrm{log}}_{3}\\left(2\\right)[\/latex]<\/p>\n<p>Using the quotient rule:<\/p>\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{3}\\left(40\\right)-{\\mathrm{log}}_{3}\\left(2\\right)={\\mathrm{log}}_{3}\\left(\\frac{40}{2}\\right)={\\mathrm{log}}_{3}\\left(20\\right)[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox tryIt\">\n<p>[ohm_question hide_question_numbers=1]129766[\/ohm_question]<\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Use rules for multiplying, raising to a power, and dividing to simplify logarithmic expressions<\/li>\n<\/ul>\n<\/section>\n<p>Logarithms are essential tools in calculus, particularly when dealing with integrals and exponential functions. Mastering how to condense logarithmic expressions using fundamental rules will simplify complex calculations and enhance your understanding of these concepts.<\/p>\n<h2>Simplify Logarithmic Expressions<\/h2>\n<p>Simplifying logarithmic expressions involves applying specific rules to transform intricate terms into more manageable ones. This process is crucial for effectively solving equations in calculus and other mathematical disciplines.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>rules for logarithms<\/h3>\n<ul>\n<li><strong>Product Rule:<\/strong> The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.\n<div style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left(M \\times N\\right)={\\mathrm{log}}_{b}\\left(M\\right)+{\\mathrm{log}}_{b}\\left(N\\right)\\text{ for }b>0[\/latex]<\/div>\n<\/li>\n<li><strong>Quotient Rule:<\/strong> The quotient rule for logarithms can be used to simplify a logarithm of a quotient by rewriting it as the difference of individual logarithms.\n<div style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left(\\frac{M}{N}\\right)={\\mathrm{log}}_{b}M-{\\mathrm{log}}_{b}N[\/latex]<\/div>\n<\/li>\n<li><strong>Power Rule:<\/strong> The power rule for logarithms 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.\n<div style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left({M}^{n}\\right)=n{\\mathrm{log}}_{b}M[\/latex]<\/div>\n<\/li>\n<\/ul>\n<\/section>\n<p>We can use the rules of logarithms to condense sums, differences, and products with the same base as a single logarithm. It is important to remember that the logarithms must have the same base to be combined.<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Given a sum, difference, or product of logarithms with the same base, write an equivalent expression as a single logarithm<\/strong><\/p>\n<ol>\n<li><strong>Apply the Power Rule:<\/strong> Identify and rewrite any terms that are powers of factors as the logarithm of a power.<\/li>\n<li><strong>Apply the Product and Quotient Rules:<\/strong> From left to right, apply the product and quotient properties to rewrite sums of logarithms as the logarithm of a product and differences as the logarithm of a quotient.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">\n<p>Use the power rule for logs to rewrite [latex]4\\mathrm{ln}\\left(x\\right)[\/latex] as a single logarithm with a leading coefficient of 1.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q959478\">Show Solution<\/button><\/p>\n<div id=\"q959478\" class=\"hidden-answer\" style=\"display: none\">\n<p>Because the logarithm of a power is the product of the exponent times the logarithm of the base, it follows that the product of a number and a logarithm can be written as a power. <\/p>\n<p>For the expression [latex]4\\mathrm{ln}\\left(x\\right)[\/latex], we identify the factor, [latex]4[\/latex], as the exponent and the argument, [latex]x[\/latex], as the base and rewrite the product as a logarithm of a power:<\/p>\n<p style=\"text-align: center;\">[latex]4\\mathrm{ln}\\left(x\\right)=\\mathrm{ln}\\left({x}^{4}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Use the power rule for logs to rewrite [latex]2{\\mathrm{log}}_{3}4[\/latex] as a single logarithm with a leading coefficient of 1.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q720709\">Show Solution<\/button><\/p>\n<div id=\"q720709\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{\\mathrm{log}}_{3}16[\/latex]<\/p><\/div>\n<\/div>\n<\/section>\n<p>In our next few examples, we will use a combination of logarithm rules to condense logarithms.<\/p>\n<section class=\"textbox example\">\n<p>Write [latex]{\\mathrm{log}}_{3}\\left(5\\right)+{\\mathrm{log}}_{3}\\left(8\\right)-{\\mathrm{log}}_{3}\\left(2\\right)[\/latex] as a single logarithm.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q484876\">Show Solution<\/button><\/p>\n<div id=\"q484876\" class=\"hidden-answer\" style=\"display: none\">\n<p>From left to right, since we have the addition of two logs, we first use the product rule:<\/p>\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{3}\\left(5\\right)+{\\mathrm{log}}_{3}\\left(8\\right)={\\mathrm{log}}_{3}\\left(5\\cdot 8\\right)={\\mathrm{log}}_{3}\\left(40\\right)[\/latex]<\/p>\n<p>This simplifies our original expression to:<\/p>\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{3}\\left(40\\right)-{\\mathrm{log}}_{3}\\left(2\\right)[\/latex]<\/p>\n<p>Using the quotient rule:<\/p>\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{3}\\left(40\\right)-{\\mathrm{log}}_{3}\\left(2\\right)={\\mathrm{log}}_{3}\\left(\\frac{40}{2}\\right)={\\mathrm{log}}_{3}\\left(20\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm129766\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=129766&theme=lumen&iframe_resize_id=ohm129766&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br 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