{"id":1709,"date":"2024-04-24T17:05:54","date_gmt":"2024-04-24T17:05:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1709"},"modified":"2024-08-05T13:02:33","modified_gmt":"2024-08-05T13:02:33","slug":"contextual-applications-of-derivatives-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/contextual-applications-of-derivatives-background-youll-need-2\/","title":{"raw":"Contextual Applications of Derivatives: Background You'll Need 2","rendered":"Contextual Applications of Derivatives: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use the power rule to simplify logarithmic expressions&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:769,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:4,&quot;12&quot;:0}\">Use the power rule to simplify logarithmic expressions<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Using the Power Rule for Logarithms<\/h2>\r\n<p>The power rule for logarithms is a fundamental concept that simplifies the process of working with logarithmic expressions involving powers.<\/p>\r\n<section class=\"textbox example\">\r\n<p>How can we take the logarithm of a power, such as [latex]{x}^{2}[\/latex]? One method is as follows:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{\\mathrm{log}}_{b}\\left({x}^{2}\\right)\\hfill &amp; ={\\mathrm{log}}_{b}\\left(x\\cdot x\\right)\\hfill \\\\ \\hfill &amp; ={\\mathrm{log}}_{b}x+{\\mathrm{log}}_{b}x\\hfill \\\\ \\hfill &amp; =2{\\mathrm{log}}_{b}x\\hfill \\end{array}[\/latex]<\/p>\r\n<\/section>\r\n<p>Notice that we used the [pb_glossary id=\"1878\"]product rule for logarithms[\/pb_glossary] to find a solution for the example above. By doing so, we have derived the <strong>power rule for logarithms<\/strong>, which says that the log of a power is equal to the exponent times the log of the base. Keep in mind that although the input to a logarithm may not be written as a power, we may be able to change it to a power.\u00a0<\/p>\r\n<section class=\"textbox example\">\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}100={10}^{2}, \\hfill &amp; \\sqrt{3}={3}^{\\frac{1}{2}}, \\hfill &amp; \\frac{1}{e}={e}^{-1}\\hfill \\end{array}[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>power rule for logarithms<\/h3>\r\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>\r\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left({M}^{n}\\right)=n{\\mathrm{log}}_{b}M[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Rewrite [latex]{\\mathrm{log}}_{2}{x}^{5}[\/latex].<\/p>\r\n<p>[reveal-answer q=\"979765\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"979765\"]<\/p>\r\n<p>The argument is already written as a power, so we identify the exponent, [latex]5[\/latex], and the base, [latex]x[\/latex], and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.<\/p>\r\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{2}\\left({x}^{5}\\right)=5{\\mathrm{log}}_{2}x[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Rewrite [latex]{\\mathrm{log}}_{3}\\left(25\\right)[\/latex] using the power rule for logs.<\/p>\r\n<p>[reveal-answer q=\"984289\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"984289\"]<\/p>\r\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>\r\n<p>Next we identify the exponent, [latex]2[\/latex], and the base, [latex]5[\/latex], and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.<\/p>\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<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]288399[\/ohm_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use the power rule to simplify logarithmic expressions&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:769,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:4,&quot;12&quot;:0}\">Use the power rule to simplify logarithmic expressions<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Using the Power Rule for Logarithms<\/h2>\n<p>The power rule for logarithms is a fundamental concept that simplifies the process of working with logarithmic expressions involving powers.<\/p>\n<section class=\"textbox example\">\n<p>How can we take the logarithm of a power, such as [latex]{x}^{2}[\/latex]? One method is as follows:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{\\mathrm{log}}_{b}\\left({x}^{2}\\right)\\hfill & ={\\mathrm{log}}_{b}\\left(x\\cdot x\\right)\\hfill \\\\ \\hfill & ={\\mathrm{log}}_{b}x+{\\mathrm{log}}_{b}x\\hfill \\\\ \\hfill & =2{\\mathrm{log}}_{b}x\\hfill \\end{array}[\/latex]<\/p>\n<\/section>\n<p>Notice that we used the <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_1709_1878\">product rule for logarithms<\/a> to find a solution for the example above. By doing so, we have derived the <strong>power rule for logarithms<\/strong>, which says that the log of a power is equal to the exponent times the log of the base. Keep in mind that although the input to a logarithm may not be written as a power, we may be able to change it to a power.\u00a0<\/p>\n<section class=\"textbox example\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}100={10}^{2}, \\hfill & \\sqrt{3}={3}^{\\frac{1}{2}}, \\hfill & \\frac{1}{e}={e}^{-1}\\hfill \\end{array}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>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<\/section>\n<section class=\"textbox example\">\n<p>Rewrite [latex]{\\mathrm{log}}_{2}{x}^{5}[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q979765\">Show Solution<\/button><\/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, [latex]5[\/latex], and the base, [latex]x[\/latex], 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<\/section>\n<section class=\"textbox example\">\n<p>Rewrite [latex]{\\mathrm{log}}_{3}\\left(25\\right)[\/latex] using the power rule for logs.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q984289\">Show Solution<\/button><\/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, [latex]2[\/latex], and the base, [latex]5[\/latex], 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<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm288399\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288399&theme=lumen&iframe_resize_id=ohm288399&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_1709_1878\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_1709_1878\"><div tabindex=\"-1\"><p>The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.<\/p>\n<p>[latex]{\\mathrm{log}}_{b}\\left(MN\\right)={\\mathrm{log}}_{b}\\left(M\\right)+{\\mathrm{log}}_{b}\\left(N\\right)\\text{ for }b>0[\/latex]<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":15,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":292,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1709"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1709\/revisions"}],"predecessor-version":[{"id":4544,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1709\/revisions\/4544"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/292"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1709\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1709"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1709"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1709"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1709"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}