{"id":282,"date":"2023-06-21T13:22:51","date_gmt":"2023-06-21T13:22:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/logarithm-rules\/"},"modified":"2023-07-04T04:23:21","modified_gmt":"2023-07-04T04:23:21","slug":"logarithm-rules","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/logarithm-rules\/","title":{"raw":"\u25aa   Properties of Logarithms","rendered":"\u25aa   Properties of Logarithms"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Rewrite a logarithmic expression using the power rule, product rule, or quotient rule.<\/li>\r\n \t<li>Expand logarithmic expressions using a combination of logarithm rules.<\/li>\r\n \t<li>Condense logarithmic expressions using logarithm rules.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Properties of Logarithms<\/h2>\r\nRecall that the logarithmic and exponential functions \"undo\" each other. This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here. First, the following properties are easy to prove.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{\\mathrm{log}}_{b}1=0\\\\{\\mathrm{log}}_{b}b=1\\end{array}[\/latex]<\/p>\r\nFor example, [latex]{\\mathrm{log}}_{5}1=0[\/latex] since [latex]{5}^{0}=1[\/latex] and [latex]{\\mathrm{log}}_{5}5=1[\/latex] since [latex]{5}^{1}=5[\/latex].\r\n\r\nNext, we have the inverse property.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\hfill \\\\ {\\mathrm{log}}_{b}\\left({b}^{x}\\right)=x\\hfill \\\\ \\text{ }{b}^{{\\mathrm{log}}_{b}x}=x,x&gt;0\\hfill \\end{array}[\/latex]<\/p>\r\nFor example, to evaluate [latex]\\mathrm{log}\\left(100\\right)[\/latex], we can rewrite the logarithm as [latex]{\\mathrm{log}}_{10}\\left({10}^{2}\\right)[\/latex] and then apply the inverse property [latex]{\\mathrm{log}}_{b}\\left({b}^{x}\\right)=x[\/latex] to get [latex]{\\mathrm{log}}_{10}\\left({10}^{2}\\right)=2[\/latex].\r\n\r\nTo evaluate [latex]{e}^{\\mathrm{ln}\\left(7\\right)}[\/latex], we can rewrite the logarithm as [latex]{e}^{{\\mathrm{log}}_{e}7}[\/latex] and then apply the inverse property [latex]{b}^{{\\mathrm{log}}_{b}x}=x[\/latex] to get [latex]{e}^{{\\mathrm{log}}_{e}7}=7[\/latex].\r\n\r\nFinally, we have the <strong>one-to-one<\/strong> property.\r\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}M={\\mathrm{log}}_{b}N\\text{ if and only if}\\text{ }M=N[\/latex]<\/p>\r\nWe can use the one-to-one property to solve the equation [latex]{\\mathrm{log}}_{3}\\left(3x\\right)={\\mathrm{log}}_{3}\\left(2x+5\\right)[\/latex] for <em>x<\/em>. Since the bases are the same, we can apply the one-to-one property by setting the arguments equal and solving for <em>x<\/em>:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}3x=2x+5\\hfill &amp; \\text{Set the arguments equal}\\text{.}\\hfill \\\\ x=5\\hfill &amp; \\text{Subtract 2}x\\text{.}\\hfill \\end{array}[\/latex]<\/p>\r\nBut what about the equation [latex]{\\mathrm{log}}_{3}\\left(3x\\right)+{\\mathrm{log}}_{3}\\left(2x+5\\right)=2[\/latex]? The one-to-one property does not help us in this instance. Before we can solve an equation like this, we need a method for combining\u00a0logarithms on the left side of the equation.\r\n<h2>Using the Product Rule for Logarithms<\/h2>\r\nRecall that we use the <em>product rule of exponents<\/em> to combine the product of exponents by adding: [latex]{x}^{a}{x}^{b}={x}^{a+b}[\/latex]. We have a similar property for logarithms, called the <strong>product rule for logarithms<\/strong>, which says that the logarithm of a product is equal to a sum of logarithms. Because logs are exponents and we multiply like bases, we can add the exponents. We will use the inverse property to derive the product rule below.\r\n\r\nGiven any real number <em>x<\/em>\u00a0and positive real numbers <em>M<\/em>, <em>N<\/em>, and <em>b<\/em>, where [latex]b\\ne 1[\/latex], we will show\r\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left(MN\\right)\\text{= }{\\mathrm{log}}_{b}\\left(M\\right)+{\\mathrm{log}}_{b}\\left(N\\right)[\/latex].<\/p>\r\nLet [latex]m={\\mathrm{log}}_{b}M[\/latex] and [latex]n={\\mathrm{log}}_{b}N[\/latex]. In exponential form, these equations are [latex]{b}^{m}=M[\/latex] and [latex]{b}^{n}=N[\/latex]. It follows that\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lllllllll}{\\mathrm{log}}_{b}\\left(MN\\right)\\hfill &amp; ={\\mathrm{log}}_{b}\\left({b}^{m}{b}^{n}\\right)\\hfill &amp; \\text{Substitute for }M\\text{ and }N.\\hfill \\\\ \\hfill &amp; ={\\mathrm{log}}_{b}\\left({b}^{m+n}\\right)\\hfill &amp; \\text{Apply the product rule for exponents}.\\hfill \\\\ \\hfill &amp; =m+n\\hfill &amp; \\text{Apply the inverse property of logs}.\\hfill \\\\ \\hfill &amp; ={\\mathrm{log}}_{b}\\left(M\\right)+{\\mathrm{log}}_{b}\\left(N\\right)\\hfill &amp; \\text{Substitute for }m\\text{ and }n.\\hfill \\end{array}[\/latex]<\/p>\r\n<p id=\"fs-id1165137749030\">Note that repeated applications of the product rule for logarithms allow us to simplify the logarithm of the product of any number of factors. For example, consider [latex]\\mathrm{log}_{b}(wxyz)[\/latex].\u00a0Using the product rule for logarithms, we can rewrite this logarithm of a product as the sum of logarithms of its factors:<\/p>\r\n\r\n<div id=\"fs-id1165137891324\" class=\"ui-has-child-title\" style=\"text-align: center;\" data-type=\"note\">[latex]\\mathrm{log}_{b}(wxyz)=\\mathrm{log}_{b}w+\\mathrm{log}_{b}x+\\mathrm{log}_{b}y+\\mathrm{log}_{b}z[\/latex]<\/div>\r\n<div class=\"textbox\">\r\n<h3>A General Note: 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)\\text{ for }b&gt;0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Product Rule for Logarithms<\/h3>\r\nExpand [latex]{\\mathrm{log}}_{3}\\left(30x\\left(3x+4\\right)\\right)[\/latex].\r\n\r\n[reveal-answer q=\"603492\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"603492\"]\r\n\r\nWe begin by writing\u00a0an equal\u00a0equation by summing the logarithms of each factor.\r\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{3}\\left(30x\\left(3x+4\\right)\\right)={\\mathrm{log}}_{3}\\left(30x\\right)+{\\mathrm{log}}_{3}\\left(3x+4\\right)={\\mathrm{log}}_{3}\\left(30\\right)+{\\mathrm{log}}_{3}\\left(x\\right)+{\\mathrm{log}}_{3}\\left(3x+4\\right)[\/latex]<\/p>\r\n<p style=\"text-align: left;\">The final expansion looks like this. Note how the factor [latex]30x[\/latex] can be expanded into the sum of two logarithms:<\/p>\r\n<p style=\"text-align: left;\">[latex]{\\mathrm{log}}_{3}\\left(30\\right)+{\\mathrm{log}}_{3}\\left(x\\right)+{\\mathrm{log}}_{3}\\left(3x+4\\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\nExpand [latex]{\\mathrm{log}}_{b}\\left(8k\\right)[\/latex].\r\n\r\n[reveal-answer q=\"829261\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"829261\"]\r\n\r\n[latex]{\\mathrm{log}}_{b}8+{\\mathrm{log}}_{b}k[\/latex][\/hidden-answer]\r\n\r\n[ohm_question]63342[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Using the Quotient Rule for Logarithms<\/h2>\r\nFor quotients, we have a similar rule for logarithms. Recall that we use the <em>quotient rule of exponents<\/em> to combine the quotient of exponents by subtracting: [latex]{x}^{\\frac{a}{b}}={x}^{a-b}[\/latex]. The <strong>quotient rule for logarithms<\/strong> says that the logarithm of a quotient is equal to a difference of logarithms. Just as with the product rule, we can use the inverse property to derive the quotient rule.\r\n\r\nGiven any real number <em>x\u00a0<\/em>and positive real numbers <em>M<\/em>, <em>N<\/em>, and <em>b<\/em>, where [latex]b\\ne 1[\/latex], we will show\r\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left(\\frac{M}{N}\\right)\\text{= }{\\mathrm{log}}_{b}\\left(M\\right)-{\\mathrm{log}}_{b}\\left(N\\right)[\/latex].<\/p>\r\nLet [latex]m={\\mathrm{log}}_{b}M[\/latex] and [latex]n={\\mathrm{log}}_{b}N[\/latex]. In exponential form, these equations are [latex]{b}^{m}=M[\/latex] and [latex]{b}^{n}=N[\/latex]. It follows that\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{\\mathrm{log}}_{b}\\left(\\frac{M}{N}\\right)\\hfill &amp; ={\\mathrm{log}}_{b}\\left(\\frac{{b}^{m}}{{b}^{n}}\\right)\\hfill &amp; \\text{Substitute for }M\\text{ and }N.\\hfill \\\\ \\hfill &amp; ={\\mathrm{log}}_{b}\\left({b}^{m-n}\\right)\\hfill &amp; \\text{Apply the quotient rule for exponents}.\\hfill \\\\ \\hfill &amp; =m-n\\hfill &amp; \\text{Apply the inverse property of logs}.\\hfill \\\\ \\hfill &amp; ={\\mathrm{log}}_{b}\\left(M\\right)-{\\mathrm{log}}_{b}\\left(N\\right)\\hfill &amp; \\text{Substitute for }m\\text{ and }n.\\hfill \\end{array}[\/latex]<\/p>\r\nFor example, to expand [latex]\\mathrm{log}\\left(\\frac{2{x}^{2}+6x}{3x+9}\\right)[\/latex], we must first express the quotient in lowest terms. Factoring and canceling, we get\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lllll}\\mathrm{log}\\left(\\frac{2{x}^{2}+6x}{3x+9}\\right) &amp; =\\mathrm{log}\\left(\\frac{2x\\left(x+3\\right)}{3\\left(x+3\\right)}\\right)\\hfill &amp; \\text{Factor the numerator and denominator}.\\hfill \\\\ &amp; \\text{}=\\mathrm{log}\\left(\\frac{2x}{3}\\right)\\hfill &amp; \\text{Cancel the common factors}.\\hfill \\end{array}[\/latex]<\/p>\r\nNext we apply the quotient rule by subtracting the logarithm of the denominator from the logarithm of the numerator. Then we apply the product rule.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}\\mathrm{log}\\left(\\frac{2x}{3}\\right) &amp; =\\mathrm{log}\\left(2x\\right)-\\mathrm{log}\\left(3\\right)\\hfill \\\\ \\text{} &amp; =\\mathrm{log}\\left(2\\right)+\\mathrm{log}\\left(x\\right)-\\mathrm{log}\\left(3\\right)\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<div class=\"textbox\">\r\n<h3>A General Note: The 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<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left(\\frac{M}{N}\\right)={\\mathrm{log}}_{b}M-{\\mathrm{log}}_{b}N[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given the logarithm of a quotient, use the quotient rule of logarithms to write an equivalent difference of logarithms<\/h3>\r\n<ol>\r\n \t<li>Express the argument in lowest terms by factoring the numerator and denominator and canceling common terms.<\/li>\r\n \t<li>Write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the numerator.<\/li>\r\n \t<li>Check to see that each term is fully expanded. If not, apply the product rule for logarithms to expand completely.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Quotient Rule for Logarithms<\/h3>\r\nExpand [latex]{\\mathrm{log}}_{2}\\left(\\frac{15x\\left(x - 1\\right)}{\\left(3x+4\\right)\\left(2-x\\right)}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"582435\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"582435\"]\r\n\r\nFirst we note that the quotient is factored and in lowest terms, so we apply the quotient rule.\r\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{2}\\left(\\frac{15x\\left(x - 1\\right)}{\\left(3x+4\\right)\\left(2-x\\right)}\\right)={\\mathrm{log}}_{2}\\left(15x\\left(x - 1\\right)\\right)-{\\mathrm{log}}_{2}\\left(\\left(3x+4\\right)\\left(2-x\\right)\\right)[\/latex]<\/p>\r\nNotice that the resulting terms are logarithms of products. To expand completely, we apply the product rule.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{\\mathrm{log}}_{2}\\left(15x\\left(x - 1\\right)\\right)-{\\mathrm{log}}_{2}\\left(\\left(3x+4\\right)\\left(2-x\\right)\\right) \\\\\\text{}= \\left[{\\mathrm{log}}_{2}\\left(15\\right)+{\\mathrm{log}}_{2}\\left(x\\right)+{\\mathrm{log}}_{2}\\left(x - 1\\right)\\right]-\\left[{\\mathrm{log}}_{2}\\left(3x+4\\right)+{\\mathrm{log}}_{2}\\left(2-x\\right)\\right]\\hfill \\\\ \\text{}={\\mathrm{log}}_{2}\\left(15\\right)+{\\mathrm{log}}_{2}\\left(x\\right)+{\\mathrm{log}}_{2}\\left(x - 1\\right)-{\\mathrm{log}}_{2}\\left(3x+4\\right)-{\\mathrm{log}}_{2}\\left(2-x\\right)\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nThere are exceptions to consider in this and later examples. First, because denominators must never be zero, this expression is not defined for [latex]x=-\\frac{4}{3}[\/latex] and <em>x\u00a0<\/em>= 2. Also, since the argument of a logarithm must be positive, we note as we observe the expanded logarithm that <em>x\u00a0<\/em>&gt; 0, <em>x\u00a0<\/em>&gt; 1, [latex]x&gt;-\\frac{4}{3}[\/latex], and <em>x\u00a0<\/em>&lt; 2. Combining these conditions is beyond the scope of this section, and we will not consider them here or in subsequent exercises.\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}}_{3}\\left(\\frac{7{x}^{2}+21x}{7x\\left(x - 1\\right)\\left(x - 2\\right)}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"442008\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"442008\"]\r\n\r\n[latex]{\\mathrm{log}}_{3}\\left(x+3\\right)-{\\mathrm{log}}_{3}\\left(x - 1\\right)-{\\mathrm{log}}_{3}\\left(x - 2\\right)[\/latex][\/hidden-answer]\r\n\r\n[ohm_question]129646[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Using the Power Rule for Logarithms<\/h2>\r\n<div class=\"textbox examples\">\r\n<h3>recall rules for rewriting exponents<\/h3>\r\n<div>The rules and properties of exponents will appear frequently when manipulating logarithms.<\/div>\r\n&nbsp;\r\n<div style=\"padding-left: 30px;\">Product Rule [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/div>\r\n<div><\/div>\r\n<div style=\"padding-left: 30px;\">Quotient Rule [latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]<\/div>\r\n<div><\/div>\r\n<div style=\"padding-left: 30px;\">Power Rule [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/div>\r\n<div><\/div>\r\n<div style=\"padding-left: 30px;\">Zero Exponent [latex]{a}^{0}=1[\/latex]<\/div>\r\n<div><\/div>\r\n<div style=\"padding-left: 30px;\">Negative Exponent [latex]{a}^{-n}=\\dfrac{1}{{a}^{n}} \\text{ and } {a}^{n}=\\dfrac{1}{{a}^{-n}}[\/latex]<\/div>\r\n<div><\/div>\r\n<div style=\"padding-left: 30px;\">Power of a Product [latex]\\large{\\left(ab\\right)}^{n}={a}^{n}{b}^{n}[\/latex]<\/div>\r\n<div><\/div>\r\n<div style=\"padding-left: 30px;\">Power of a Quotient [latex]\\large{\\left(\\dfrac{a}{b}\\right)}^{n}=\\dfrac{{a}^{n}}{{b}^{n}}[\/latex]<\/div>\r\n<\/div>\r\nWe have explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as [latex]{x}^{2}[\/latex]? One method is as follows:\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\nNotice that we used the <strong>product rule for logarithms<\/strong> 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. For 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\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\">\r\n<h3>How To: Given the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm<\/h3>\r\n<ol>\r\n \t<li>Express the argument as a power, if needed.<\/li>\r\n \t<li>Write the equivalent expression by multiplying the exponent times the logarithm of the base.<\/li>\r\n<\/ol>\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[ohm_question]16926[\/ohm_question]\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[ohm_question]63350[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Rewrite a logarithmic expression using the power rule, product rule, or quotient rule.<\/li>\n<li>Expand logarithmic expressions using a combination of logarithm rules.<\/li>\n<li>Condense logarithmic expressions using logarithm rules.<\/li>\n<\/ul>\n<\/div>\n<h2>Properties of Logarithms<\/h2>\n<p>Recall that the logarithmic and exponential functions &#8220;undo&#8221; each other. This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here. First, the following properties are easy to prove.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{\\mathrm{log}}_{b}1=0\\\\{\\mathrm{log}}_{b}b=1\\end{array}[\/latex]<\/p>\n<p>For example, [latex]{\\mathrm{log}}_{5}1=0[\/latex] since [latex]{5}^{0}=1[\/latex] and [latex]{\\mathrm{log}}_{5}5=1[\/latex] since [latex]{5}^{1}=5[\/latex].<\/p>\n<p>Next, we have the inverse property.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\hfill \\\\ {\\mathrm{log}}_{b}\\left({b}^{x}\\right)=x\\hfill \\\\ \\text{ }{b}^{{\\mathrm{log}}_{b}x}=x,x>0\\hfill \\end{array}[\/latex]<\/p>\n<p>For example, to evaluate [latex]\\mathrm{log}\\left(100\\right)[\/latex], we can rewrite the logarithm as [latex]{\\mathrm{log}}_{10}\\left({10}^{2}\\right)[\/latex] and then apply the inverse property [latex]{\\mathrm{log}}_{b}\\left({b}^{x}\\right)=x[\/latex] to get [latex]{\\mathrm{log}}_{10}\\left({10}^{2}\\right)=2[\/latex].<\/p>\n<p>To evaluate [latex]{e}^{\\mathrm{ln}\\left(7\\right)}[\/latex], we can rewrite the logarithm as [latex]{e}^{{\\mathrm{log}}_{e}7}[\/latex] and then apply the inverse property [latex]{b}^{{\\mathrm{log}}_{b}x}=x[\/latex] to get [latex]{e}^{{\\mathrm{log}}_{e}7}=7[\/latex].<\/p>\n<p>Finally, we have the <strong>one-to-one<\/strong> property.<\/p>\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}M={\\mathrm{log}}_{b}N\\text{ if and only if}\\text{ }M=N[\/latex]<\/p>\n<p>We can use the one-to-one property to solve the equation [latex]{\\mathrm{log}}_{3}\\left(3x\\right)={\\mathrm{log}}_{3}\\left(2x+5\\right)[\/latex] for <em>x<\/em>. Since the bases are the same, we can apply the one-to-one property by setting the arguments equal and solving for <em>x<\/em>:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}3x=2x+5\\hfill & \\text{Set the arguments equal}\\text{.}\\hfill \\\\ x=5\\hfill & \\text{Subtract 2}x\\text{.}\\hfill \\end{array}[\/latex]<\/p>\n<p>But what about the equation [latex]{\\mathrm{log}}_{3}\\left(3x\\right)+{\\mathrm{log}}_{3}\\left(2x+5\\right)=2[\/latex]? The one-to-one property does not help us in this instance. Before we can solve an equation like this, we need a method for combining\u00a0logarithms on the left side of the equation.<\/p>\n<h2>Using the Product Rule for Logarithms<\/h2>\n<p>Recall that we use the <em>product rule of exponents<\/em> to combine the product of exponents by adding: [latex]{x}^{a}{x}^{b}={x}^{a+b}[\/latex]. We have a similar property for logarithms, called the <strong>product rule for logarithms<\/strong>, which says that the logarithm of a product is equal to a sum of logarithms. Because logs are exponents and we multiply like bases, we can add the exponents. We will use the inverse property to derive the product rule below.<\/p>\n<p>Given any real number <em>x<\/em>\u00a0and positive real numbers <em>M<\/em>, <em>N<\/em>, and <em>b<\/em>, where [latex]b\\ne 1[\/latex], we will show<\/p>\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left(MN\\right)\\text{= }{\\mathrm{log}}_{b}\\left(M\\right)+{\\mathrm{log}}_{b}\\left(N\\right)[\/latex].<\/p>\n<p>Let [latex]m={\\mathrm{log}}_{b}M[\/latex] and [latex]n={\\mathrm{log}}_{b}N[\/latex]. In exponential form, these equations are [latex]{b}^{m}=M[\/latex] and [latex]{b}^{n}=N[\/latex]. It follows that<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lllllllll}{\\mathrm{log}}_{b}\\left(MN\\right)\\hfill & ={\\mathrm{log}}_{b}\\left({b}^{m}{b}^{n}\\right)\\hfill & \\text{Substitute for }M\\text{ and }N.\\hfill \\\\ \\hfill & ={\\mathrm{log}}_{b}\\left({b}^{m+n}\\right)\\hfill & \\text{Apply the product rule for exponents}.\\hfill \\\\ \\hfill & =m+n\\hfill & \\text{Apply the inverse property of logs}.\\hfill \\\\ \\hfill & ={\\mathrm{log}}_{b}\\left(M\\right)+{\\mathrm{log}}_{b}\\left(N\\right)\\hfill & \\text{Substitute for }m\\text{ and }n.\\hfill \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165137749030\">Note that repeated applications of the product rule for logarithms allow us to simplify the logarithm of the product of any number of factors. For example, consider [latex]\\mathrm{log}_{b}(wxyz)[\/latex].\u00a0Using the product rule for logarithms, we can rewrite this logarithm of a product as the sum of logarithms of its factors:<\/p>\n<div id=\"fs-id1165137891324\" class=\"ui-has-child-title\" style=\"text-align: center;\" data-type=\"note\">[latex]\\mathrm{log}_{b}(wxyz)=\\mathrm{log}_{b}w+\\mathrm{log}_{b}x+\\mathrm{log}_{b}y+\\mathrm{log}_{b}z[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>A General Note: 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)\\text{ for }b>0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Product Rule for Logarithms<\/h3>\n<p>Expand [latex]{\\mathrm{log}}_{3}\\left(30x\\left(3x+4\\right)\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q603492\">Show Solution<\/span><\/p>\n<div id=\"q603492\" class=\"hidden-answer\" style=\"display: none\">\n<p>We begin by writing\u00a0an equal\u00a0equation by summing the logarithms of each factor.<\/p>\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{3}\\left(30x\\left(3x+4\\right)\\right)={\\mathrm{log}}_{3}\\left(30x\\right)+{\\mathrm{log}}_{3}\\left(3x+4\\right)={\\mathrm{log}}_{3}\\left(30\\right)+{\\mathrm{log}}_{3}\\left(x\\right)+{\\mathrm{log}}_{3}\\left(3x+4\\right)[\/latex]<\/p>\n<p style=\"text-align: left;\">The final expansion looks like this. Note how the factor [latex]30x[\/latex] can be expanded into the sum of two logarithms:<\/p>\n<p style=\"text-align: left;\">[latex]{\\mathrm{log}}_{3}\\left(30\\right)+{\\mathrm{log}}_{3}\\left(x\\right)+{\\mathrm{log}}_{3}\\left(3x+4\\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}}_{b}\\left(8k\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q829261\">Show Solution<\/span><\/p>\n<div id=\"q829261\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{\\mathrm{log}}_{b}8+{\\mathrm{log}}_{b}k[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm63342\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=63342&theme=oea&iframe_resize_id=ohm63342&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Using the Quotient Rule for Logarithms<\/h2>\n<p>For quotients, we have a similar rule for logarithms. Recall that we use the <em>quotient rule of exponents<\/em> to combine the quotient of exponents by subtracting: [latex]{x}^{\\frac{a}{b}}={x}^{a-b}[\/latex]. The <strong>quotient rule for logarithms<\/strong> says that the logarithm of a quotient is equal to a difference of logarithms. Just as with the product rule, we can use the inverse property to derive the quotient rule.<\/p>\n<p>Given any real number <em>x\u00a0<\/em>and positive real numbers <em>M<\/em>, <em>N<\/em>, and <em>b<\/em>, where [latex]b\\ne 1[\/latex], we will show<\/p>\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left(\\frac{M}{N}\\right)\\text{= }{\\mathrm{log}}_{b}\\left(M\\right)-{\\mathrm{log}}_{b}\\left(N\\right)[\/latex].<\/p>\n<p>Let [latex]m={\\mathrm{log}}_{b}M[\/latex] and [latex]n={\\mathrm{log}}_{b}N[\/latex]. In exponential form, these equations are [latex]{b}^{m}=M[\/latex] and [latex]{b}^{n}=N[\/latex]. It follows that<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{\\mathrm{log}}_{b}\\left(\\frac{M}{N}\\right)\\hfill & ={\\mathrm{log}}_{b}\\left(\\frac{{b}^{m}}{{b}^{n}}\\right)\\hfill & \\text{Substitute for }M\\text{ and }N.\\hfill \\\\ \\hfill & ={\\mathrm{log}}_{b}\\left({b}^{m-n}\\right)\\hfill & \\text{Apply the quotient rule for exponents}.\\hfill \\\\ \\hfill & =m-n\\hfill & \\text{Apply the inverse property of logs}.\\hfill \\\\ \\hfill & ={\\mathrm{log}}_{b}\\left(M\\right)-{\\mathrm{log}}_{b}\\left(N\\right)\\hfill & \\text{Substitute for }m\\text{ and }n.\\hfill \\end{array}[\/latex]<\/p>\n<p>For example, to expand [latex]\\mathrm{log}\\left(\\frac{2{x}^{2}+6x}{3x+9}\\right)[\/latex], we must first express the quotient in lowest terms. Factoring and canceling, we get<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lllll}\\mathrm{log}\\left(\\frac{2{x}^{2}+6x}{3x+9}\\right) & =\\mathrm{log}\\left(\\frac{2x\\left(x+3\\right)}{3\\left(x+3\\right)}\\right)\\hfill & \\text{Factor the numerator and denominator}.\\hfill \\\\ & \\text{}=\\mathrm{log}\\left(\\frac{2x}{3}\\right)\\hfill & \\text{Cancel the common factors}.\\hfill \\end{array}[\/latex]<\/p>\n<p>Next we apply the quotient rule by subtracting the logarithm of the denominator from the logarithm of the numerator. Then we apply the product rule.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}\\mathrm{log}\\left(\\frac{2x}{3}\\right) & =\\mathrm{log}\\left(2x\\right)-\\mathrm{log}\\left(3\\right)\\hfill \\\\ \\text{} & =\\mathrm{log}\\left(2\\right)+\\mathrm{log}\\left(x\\right)-\\mathrm{log}\\left(3\\right)\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The 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<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{b}\\left(\\frac{M}{N}\\right)={\\mathrm{log}}_{b}M-{\\mathrm{log}}_{b}N[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the logarithm of a quotient, use the quotient rule of logarithms to write an equivalent difference of logarithms<\/h3>\n<ol>\n<li>Express the argument in lowest terms by factoring the numerator and denominator and canceling common terms.<\/li>\n<li>Write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the numerator.<\/li>\n<li>Check to see that each term is fully expanded. If not, apply the product rule for logarithms to expand completely.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Quotient Rule for Logarithms<\/h3>\n<p>Expand [latex]{\\mathrm{log}}_{2}\\left(\\frac{15x\\left(x - 1\\right)}{\\left(3x+4\\right)\\left(2-x\\right)}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q582435\">Show Solution<\/span><\/p>\n<div id=\"q582435\" class=\"hidden-answer\" style=\"display: none\">\n<p>First we note that the quotient is factored and in lowest terms, so we apply the quotient rule.<\/p>\n<p style=\"text-align: center;\">[latex]{\\mathrm{log}}_{2}\\left(\\frac{15x\\left(x - 1\\right)}{\\left(3x+4\\right)\\left(2-x\\right)}\\right)={\\mathrm{log}}_{2}\\left(15x\\left(x - 1\\right)\\right)-{\\mathrm{log}}_{2}\\left(\\left(3x+4\\right)\\left(2-x\\right)\\right)[\/latex]<\/p>\n<p>Notice that the resulting terms are logarithms of products. To expand completely, we apply the product rule.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}{\\mathrm{log}}_{2}\\left(15x\\left(x - 1\\right)\\right)-{\\mathrm{log}}_{2}\\left(\\left(3x+4\\right)\\left(2-x\\right)\\right) \\\\\\text{}= \\left[{\\mathrm{log}}_{2}\\left(15\\right)+{\\mathrm{log}}_{2}\\left(x\\right)+{\\mathrm{log}}_{2}\\left(x - 1\\right)\\right]-\\left[{\\mathrm{log}}_{2}\\left(3x+4\\right)+{\\mathrm{log}}_{2}\\left(2-x\\right)\\right]\\hfill \\\\ \\text{}={\\mathrm{log}}_{2}\\left(15\\right)+{\\mathrm{log}}_{2}\\left(x\\right)+{\\mathrm{log}}_{2}\\left(x - 1\\right)-{\\mathrm{log}}_{2}\\left(3x+4\\right)-{\\mathrm{log}}_{2}\\left(2-x\\right)\\hfill \\end{array}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>There are exceptions to consider in this and later examples. First, because denominators must never be zero, this expression is not defined for [latex]x=-\\frac{4}{3}[\/latex] and <em>x\u00a0<\/em>= 2. Also, since the argument of a logarithm must be positive, we note as we observe the expanded logarithm that <em>x\u00a0<\/em>&gt; 0, <em>x\u00a0<\/em>&gt; 1, [latex]x>-\\frac{4}{3}[\/latex], and <em>x\u00a0<\/em>&lt; 2. Combining these conditions is beyond the scope of this section, and we will not consider them here or in subsequent exercises.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Expand [latex]{\\mathrm{log}}_{3}\\left(\\frac{7{x}^{2}+21x}{7x\\left(x - 1\\right)\\left(x - 2\\right)}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q442008\">Show Solution<\/span><\/p>\n<div id=\"q442008\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{\\mathrm{log}}_{3}\\left(x+3\\right)-{\\mathrm{log}}_{3}\\left(x - 1\\right)-{\\mathrm{log}}_{3}\\left(x - 2\\right)[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm129646\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=129646&theme=oea&iframe_resize_id=ohm129646&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Using the Power Rule for Logarithms<\/h2>\n<div class=\"textbox examples\">\n<h3>recall rules for rewriting exponents<\/h3>\n<div>The rules and properties of exponents will appear frequently when manipulating logarithms.<\/div>\n<p>&nbsp;<\/p>\n<div style=\"padding-left: 30px;\">Product Rule [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/div>\n<div><\/div>\n<div style=\"padding-left: 30px;\">Quotient Rule [latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]<\/div>\n<div><\/div>\n<div style=\"padding-left: 30px;\">Power Rule [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/div>\n<div><\/div>\n<div style=\"padding-left: 30px;\">Zero Exponent [latex]{a}^{0}=1[\/latex]<\/div>\n<div><\/div>\n<div style=\"padding-left: 30px;\">Negative Exponent [latex]{a}^{-n}=\\dfrac{1}{{a}^{n}} \\text{ and } {a}^{n}=\\dfrac{1}{{a}^{-n}}[\/latex]<\/div>\n<div><\/div>\n<div style=\"padding-left: 30px;\">Power of a Product [latex]\\large{\\left(ab\\right)}^{n}={a}^{n}{b}^{n}[\/latex]<\/div>\n<div><\/div>\n<div style=\"padding-left: 30px;\">Power of a Quotient [latex]\\large{\\left(\\dfrac{a}{b}\\right)}^{n}=\\dfrac{{a}^{n}}{{b}^{n}}[\/latex]<\/div>\n<\/div>\n<p>We have explored the product rule and the quotient rule, but 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<p>Notice that we used the <strong>product rule for logarithms<\/strong> 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. For example,<\/p>\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<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\">\n<h3>How To: Given the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm<\/h3>\n<ol>\n<li>Express the argument as a power, if needed.<\/li>\n<li>Write the equivalent expression by multiplying the exponent times the logarithm of the base.<\/li>\n<\/ol>\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<p><iframe loading=\"lazy\" id=\"ohm16926\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=16926&theme=oea&iframe_resize_id=ohm16926&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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<p><iframe loading=\"lazy\" id=\"ohm63350\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=63350&theme=oea&iframe_resize_id=ohm63350&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-282\">\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 63350. <strong>Authored by<\/strong>: Brin,Leon. <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":43,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 63350\",\"author\":\"Brin,Leon\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + 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