{"id":2511,"date":"2022-06-03T20:25:25","date_gmt":"2022-06-03T20:25:25","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=2511"},"modified":"2025-12-07T22:45:56","modified_gmt":"2025-12-07T22:45:56","slug":"6-4-algebraic-analysis-on-the-properties-of-logarithms","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/6-4-algebraic-analysis-on-the-properties-of-logarithms\/","title":{"raw":"6.4: Algebraic Analysis on the Properties of Logarithms","rendered":"6.4: Algebraic Analysis on the Properties of Logarithms"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Objectives<\/h1>\r\n<ul>\r\n \t<li>Apply the product rule<\/li>\r\n \t<li>Apply the quotient rule<\/li>\r\n \t<li>Apply the power rule<\/li>\r\n \t<li>Apply the change of base rule<\/li>\r\n \t<li>Explain natural logarithms and common logarithms<\/li>\r\n<\/ul>\r\n<\/div>\r\nSInce logarithms are exponents, they have a list of rules that go along with them.\r\n<h2>The Identity Property for Logarithms<\/h2>\r\nThe definition of logarithm states that [latex]\\log_b{a}=x[\/latex] is equivalent to [latex]a=b^x[\/latex]. Consequently, since\u00a0[latex]\\log_b{x}=\\log_b{x}[\/latex] then [latex]x=b^{\\log_b{x}}[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Identity property for Logarithms<\/h3>\r\nFor all real numbers [latex]b&gt;0, b\\neq1[\/latex],\r\n<p style=\"text-align: center;\">[latex]b^{\\log_b{x}}=x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nEvaluate:\r\n<ol>\r\n \t<li>[latex]2^{\\log_2{5}}[\/latex]<\/li>\r\n \t<li>[latex]3^{\\log_3{7}}[\/latex]<\/li>\r\n \t<li>[latex]7^{\\log_7{2}}[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\nSInce\u00a0[latex]b^{\\log_b{x}}=x[\/latex],\r\n<ol>\r\n \t<li>[latex]2^{\\log_2{5}}=5[\/latex]<\/li>\r\n \t<li>[latex]3^{\\log_3{7}}=7[\/latex]<\/li>\r\n \t<li>[latex]7^{\\log_7{2}}=2[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nEvaluate:\r\n<ol>\r\n \t<li>[latex]2^{\\log_2{3}}[\/latex]<\/li>\r\n \t<li>[latex]9^{\\log_9{7}}[\/latex]<\/li>\r\n \t<li>[latex]2^{\\log_2{5}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm150\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm150\"]\r\n<ol>\r\n \t<li>3<\/li>\r\n \t<li>7<\/li>\r\n \t<li>5<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>The Product Property for Logarithms<\/h2>\r\nThe product Property for logarithms mimics the product Property for exponents. SInce logarithms are exponents the exponential property [latex]a^m\\cdot a^n=a^{m+n}[\/latex] gets translated into logarithmic form. The multiplication of terms inside the argument of a logarithm is equal to the addition of logarithms of each term.\r\n<div class=\"textbox shaded\">\r\n<h3>The Product Property for logarithms<\/h3>\r\nFor any positive real numbers [latex]b,\\;M,\\;N[\/latex], and [latex]b\\neq1[\/latex],\r\n<p style=\"text-align: center;\">[latex]\\log_b{MN} = \\log_b{M}+\\log_b{N}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nExpand the logarithm\u00a0[latex]\\log_3{3x}[\/latex].\r\n<h4>Solution<\/h4>\r\nThe argument consists of the product of 3 and [latex]x[\/latex], so applying the product property:\r\n<p style=\"text-align: center;\">[latex]\\log_3{3x}=\\log_3{3}+\\log_3{x}[\/latex]<\/p>\r\nWe can then evaluate [latex]\\log_3{3}=1[\/latex], to get:\r\n<p style=\"text-align: center;\">[latex]\\log_3{3}+\\log_3{x}=1+\\log_3x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nExpand the logarithm\u00a0[latex]\\log_5{[5s(t-3)]}[\/latex].\r\n<h4>Solution<\/h4>\r\nThe argument consists of the product of [latex]5\\cdot s\\cdot (t-3)[\/latex], so we can expand the product property:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\log_5{[5s(t-3)]} &amp;= \\log_5{5}+\\log_5{s} +\\log_5{(t-3)}\\\\&amp;=1+\\log_5{s}+\\log_5{(t-3)}\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nExpand the logarithm\u00a0[latex]\\log_4{16x}[\/latex].\r\n\r\n[reveal-answer q=\"hjm311\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm311\"]\r\n\r\n[latex]2+\\log_4x[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nExpand the logarithm\u00a0[latex]\\log_2{[8(s+2)(t-5)]}[\/latex].\r\n\r\n[reveal-answer q=\"hjm407\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm407\"]\r\n\r\n[latex]3+\\log_2(s+2)+\\log_2(t-5)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>The Quotient Property for Logarithms<\/h2>\r\nThe quotient property for logarithms mimics the quotient property for exponents. SInce logarithms are exponents the exponential property [latex]\\dfrac{a^m}{a^n}=a^{m-n}[\/latex] gets translated into logarithmic form. The division of terms inside the argument of a logarithm is equal to the subtraction of logarithms of each term.\r\n<div class=\"textbox shaded\">\r\n<h3>The quotient Property for logarithms<\/h3>\r\n<p style=\"text-align: left;\">For any positive real numbers [latex]b,\\;M,\\;N[\/latex], and [latex]b\\neq1[\/latex],<\/p>\r\n<p style=\"text-align: center;\">[latex]\\log_b{\\dfrac{M}{N}} = \\log_b{M}-\\log_b{N}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nExpand the logarithm\u00a0[latex]\\log_{10}{\\frac{x}{10}}[\/latex].\r\n<h4>Solution<\/h4>\r\nThe argument of the log consists of a quotient of the two terms\u00a0[latex]x[\/latex] and 10, so we apply the quotient property:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\log_{10}{\\frac{x}{10}} &amp;= \\log_{10}{x}-\\log_{10}{10} \\\\&amp;=\\log_{10}{x}-1\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nExpand the logarithm\u00a0[latex]\\log_{4}{\\frac{x}{4}}[\/latex].\r\n\r\n[reveal-answer q=\"hjm492\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm492\"]\r\n\r\n[latex]\\log_4x-1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>The Power Property for Logarithms<\/h2>\r\nThe power Property comes from the product rule of exponents. [latex]\\log_b{M^r}=\\log_b{(M\\cdot M\\cdot M...M\\cdot M)}=\\log_bM+\\log_bM+\\log_bM+...+\\log_bM=r\\log_bM[\/latex]. If the argument of a logarithm is a single term with an exponent, we may pull the exponent in front of the logarithm.\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: left;\">The Power Property for Logarithms<\/h3>\r\nFor any positive real numbers [latex]b,\\;M,\\;N[\/latex], and [latex]b\\neq1[\/latex],\r\n<p style=\"text-align: center;\">[latex]\\log_b{M^r}=r\\times\\log_b{M}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 5<\/h3>\r\nExpand the logarithm\u00a0[latex]\\log_{2}{x^4}[\/latex].\r\n<h4>Solution<\/h4>\r\nThe argument of the logarithm is [latex]x^4[\/latex], a single term raised to a power, so we can apply the power property of logarithms:\r\n<p style=\"text-align: center;\">[latex]\\log_{2}{x^4}=4\\log_2{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nExpand the logarithm\u00a0[latex]\\log_{7}{x^{-2}}[\/latex].\r\n\r\n[reveal-answer q=\"hjm920\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm920\"]\r\n\r\n[latex]-2\\log_7x[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Expanding a Logarithm<\/h2>\r\nThe properties we have learned so far to expand a logarithm can be used in conjunction with one another.\r\n<div class=\"textbox examples\">\r\n<h3>Example 6<\/h3>\r\nExpand the logarithm [latex]\\log_{2}{\\frac{2y}{5}}[\/latex].\r\n<h4>Solution<\/h4>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\log_{2}{\\frac{2y}{5}} &amp;=\\log_2{(2y)}-\\log_2{5}&amp;&amp;\\text{Quotient Property}\\\\&amp;=\\log_2{2}+\\log_2{y}-\\log_2{5}&amp;&amp;\\text{Product Property}\\\\&amp;=1+\\log_2{y}-\\log_2{5}&amp;&amp;\\text{IdentityProperty}\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 7<\/h3>\r\nExpand the logarithm [latex]\\log_{4}{\\frac{16y^3}{5}}[\/latex].\r\n<h4>Solution<\/h4>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\log_{4}{\\frac{16y^3}{5}} &amp;=\\log_4{(16y^3)}-\\log_4{5}&amp;&amp;\\text{Quotient Property}\\\\&amp;=\\log_4{16}+\\log_4{y^3}-\\log_4{5}&amp;&amp;\\text{Product Property}\\\\&amp;=\\log_4{4^2}+3\\log_4{y}-\\log_4{5}&amp;&amp;\\text{Power Property}\\\\&amp;=2+3\\log_4{y}-\\log_4{5}&amp;&amp;\\text{Identity Property}\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 6<\/h3>\r\nExpand the logarithm [latex]\\log_{10}{100x^3y^5}[\/latex].\r\n\r\n[reveal-answer q=\"hjm882\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm882\"]\r\n\r\n[latex]2+3\\log_{10}x+5\\log_{10}y[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 7<\/h3>\r\nExpand the logarithm [latex]\\log_{2}{\\dfrac{32x}{y^5}}[\/latex].\r\n\r\n[reveal-answer q=\"hjm376\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm376\"]\r\n\r\n[latex]5+\\log_2x-5\\log_2y[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Combining Logarithms<\/h2>\r\nThe properties for exponents we applied to expand a logarithm can be also be used (in reverse) to combine logarithms into a single logarithm. The assumption is that these logarithms have the same base.\r\n<div class=\"textbox examples\">\r\n<h3>Example 8<\/h3>\r\nCombine [latex]\\log_4{2}+\\log_4{10}-log_4{5}[\/latex] into a single logarithm.\r\n<h4>Solution<\/h4>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\log_4{2}+\\log_4{10}-\\log_4{5}&amp;=\\log_4{(2\\times10)}-\\log_4{5}&amp;&amp;\\text{Product Property}\\\\&amp;=\\log_4{\\frac{20}{5}}&amp;&amp;\\text{Quotient Property}\\\\&amp;=\\log_4{4}&amp;&amp;\\text{Simplify}\\\\&amp;=1\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 9<\/h3>\r\nCombine [latex]2\\log_2{(x-2)}-\\log_2{(x^2-x-2)}[\/latex] into a single logarithm.\r\n<h4>Solution<\/h4>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}2\\log_2{(x-2)}-\\log_2{(x^2-x-2)}&amp;=\\log_2{(x-2)^2}-\\log_2{(x^2-x-2)}&amp;&amp;\\text{Power Property}\\\\&amp;=\\log_2{\\frac{(x-2)^2}{x^2-x-2}}&amp;&amp;\\text{Quotient Property}\\\\&amp;=\\log_2{\\frac{(x-2)^2}{(x-2)(x+1)}}&amp;&amp;\\text{Factor}\\\\&amp;=\\log_2{\\frac{(x-2)}{(x+1)}}&amp;&amp;\\text{Simplify}\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 8<\/h3>\r\nCombine [latex]\\log_2{3}+4\\log_2{x}+5\\log_2{y}[\/latex] into a single logarithm.\r\n\r\n[reveal-answer q=\"hjm202\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm202\"]\r\n\r\n[latex]\\log_2{3x^4y^5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 9<\/h3>\r\nCombine [latex]4\\log_2{x}-\\log_2{7}-5\\log_2{y}[\/latex] into a single logarithm.\r\n\r\n[reveal-answer q=\"hjm200\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm200\"]\r\n\r\n[latex]\\log_2{\\dfrac{x^4}{7y^5}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 10<\/h3>\r\nCombine [latex]2\\log_2{(x-1)}-\\log_2{(x^2-1)}[\/latex] into a single logarithm.\r\n\r\n[reveal-answer q=\"hjm201\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm201\"]\r\n\r\n[latex]\\log_2{\\dfrac{x-1}{x+1}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Common and Natural Logarithms<\/h2>\r\nIf the base of a logarithm is 10, the logarithm is call a <em><strong>common logarithm<\/strong><\/em>. This is because the common number system we use in our daily life is base-10. If the base of a logarithm is the number [latex]e[\/latex] (also known as Euler's number), the logarithm is call a <strong><em>natural logarithm<\/em><\/strong>. This is because the number [latex]e[\/latex] is an irrational number equal to 2.718281828459...; the ellipses ... mean that the decimal never ends and never repeats. The number [latex]e[\/latex] was\u00a0first discovered in 1683 by the Swiss mathematician Jacob Bernoulli but was first evaluated by Leonhard Euler in 1737. Consequently, it is often referred to as Euler's number.\u00a0The\u00a0natural\u00a0logarithm is one of the most useful functions in\u00a0mathematics, with applications\u00a0throughout the\u00a0physical\u00a0and biological sciences.\r\n\r\n<img class=\"alignleft wp-image-3047\" style=\"orphans: 1;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/28010755\/log-and-ln-calaulator-195x300.png\" alt=\"Calculator buttons\" width=\"116\" height=\"179\" \/>\r\n\r\n&nbsp;\r\n\r\nMost scientific calculators have both [latex]\\log_{10}[\/latex], which is written as [latex]\\log[\/latex] (the base is so <em>common<\/em> it is omitted) and [latex]\\log_e[\/latex] which is written [latex]\\ln[\/latex] ([latex]\\ln[\/latex] refers to\u00a0<em>logarithm naturale).<\/em>\r\n\r\nNotice that behind the log button is [latex]10^x[\/latex], and behind the ln button is [latex]e^x[\/latex]. These are the inverse functions.\r\n\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>Example 10<\/h3>\r\nUse a calculator to evaluate the logarithms exactly or to 5 decimal places:\r\n<ol>\r\n \t<li>[latex]\\log4[\/latex]<\/li>\r\n \t<li>[latex]\\log100[\/latex]<\/li>\r\n \t<li>[latex]\\ln4[\/latex]<\/li>\r\n \t<li>[latex]\\ln{e}[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>[latex]\\log4=0.60206[\/latex]<\/li>\r\n \t<li>[latex]\\log100=2[\/latex]<\/li>\r\n \t<li>[latex]\\ln{4}=1.38629[\/latex]<\/li>\r\n \t<li>[latex]\\ln{e}=1[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\nAll of the properties of logarithms apply to common and natural logarithms.\r\n<div class=\"textbox shaded\">\r\n<h3>Properties of Logarithms<\/h3>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%;\">Property<\/th>\r\n<th style=\"width: 33.3333%;\">Base 10<\/th>\r\n<th style=\"width: 33.3333%;\">Base [latex]e[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">Identity<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]\\log{10}=1[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]\\ln{e}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">Product<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]\\log{MN}=\\log{M}+\\log{N}[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]\\ln{MN}=\\ln{M}+\\ln{N}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">Quotient<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]\\log{\\dfrac{M}{N}}=\\log{M}-\\log{N}[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]\\ln{\\dfrac{M}{N}}=\\ln{M}-\\ln{N}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">Power<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]\\log{M^r}=r\\log{M}[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]\\ln{M^r}=r\\ln{M}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 11<\/h3>\r\nEvaluate [latex]\\ln{e^2}[\/latex].\r\n<h4>Solution<\/h4>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\ln{e^2}&amp;=2\\ln{e}\\\\&amp;=2\\times1\\\\&amp;=2\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 11<\/h3>\r\nEvaluate [latex]\\log{10,000}[\/latex].\r\n\r\n[reveal-answer q=\"hjm869\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm869\"][latex]4[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 12<\/h3>\r\nSimplify to a single logarithm: [latex]\\log{(x^2+4x+3)}-2\\log{(x+1)}[\/latex]\r\n<h4>Solution<\/h4>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\log{(x^2+4x+3)}-2\\log{(x+1)}&amp;=\\log{(x+3)(x+1)}-\\log{(x+1)^2}&amp;&amp;\\text{Factor. Power property.}\\\\&amp;=\\log{\\dfrac{(x+3)(x+1)}{(x+1)^2}}&amp;&amp;\\text{Quotient property}\\\\&amp;=\\log{\\dfrac{x+3}{x+1}}&amp;&amp;\\text{Simplify}\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 12<\/h3>\r\nSimplify to a single logarithm:\u00a0[latex]4\\ln{x}-2\\ln{(x+1)}+\\ln{(x^2-1)}[\/latex]\r\n\r\n[reveal-answer q=\"hjm751\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm751\"][latex]\\ln{\\dfrac{x^4(x-1)}{(x+1)}}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>The Change of Base Rule<\/h2>\r\nCalculators are a great way to evaluate logarithms with base 10 or [latex]e[\/latex]. But what if we want to evaluate a logarithm with a different base? The good news is that we can change the base of any logarithm to any base we wish to use, most importantly base 10 and [latex]e[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>The Change of Base Formula<\/h3>\r\nFor any positive real number [latex]m[\/latex] with [latex]m\\neq1[\/latex],\r\n<p style=\"text-align: center;\">[latex]\\log_b{a}=\\dfrac{\\log_m{a}}{\\log_m{b}}[\/latex]<\/p>\r\n\r\n<\/div>\r\nSInce we usually have access to a calculator with logarithms of base 10 and [latex]e[\/latex], it is important to be able to change the base of any logarithm to 10 or\u00a0[latex]e[\/latex]. The change of base formula can be used:\r\n<p style=\"text-align: center;\">[latex]\\log_ba=\\dfrac{\\log{a}}{\\log{b}}[\/latex]\u00a0 \u00a0 and\u00a0 \u00a0\u00a0[latex]\\log_ba=\\dfrac{\\ln{a}}{\\ln{b}}[\/latex]<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 13<\/h3>\r\nEvaluate\u00a0[latex]\\log_2{5}[\/latex] to 5 decimal places.\r\n<h4>Solution<\/h4>\r\nConverting to common logs:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\log_2{5}&amp;=\\dfrac{\\log{5}}{\\log{2}}\\\\&amp;=2.32193\\end{aligned}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">or<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\log_2{5}&amp;=\\dfrac{\\ln{5}}{\\ln{2}}\\\\&amp;=2.32193\\end{aligned}[\/latex]<\/p>\r\nNotice that when using a calculator we never write the answer to the numerator and denominator of a fraction, then divide those two rounded numbers. This is because rounding each number than dividing brings in rounding error. Instead we complete the full calculation in the calculator, then round at the very end.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 13<\/h3>\r\nEvaluate [latex]\\log_9{32}[\/latex] to 5 decimal places.\r\n\r\n[reveal-answer q=\"hjm325\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm325\"][latex]1.57732[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Composition with the Inverse Function (Application of the Properties of Logarithms)<\/h2>\r\nIn chapter 3.3.2, we learned that [latex]f(x)[\/latex] and [latex]g(x)[\/latex] are inverses of each other if\u00a0[latex]f\\left(g\\left(x\\right)\\right)=x[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)=x[\/latex]. We may use this property to determine if the inverse function we found is correct or not. For example, suppose someone found that the inverse function of [latex]f(x)=\\log_2{(x+3)}-5[\/latex] is [latex]f^{-1}(x)=2^{x+5}-3[\/latex]. We may justify this answer by checking if [latex]f(f^{-1}(x))=x[\/latex] and\u00a0[latex]f^{-1}(f(x))=x[\/latex]. If either one is false, then the inverse function is incorrect.\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}f(f^{-1}(x))&amp;=\\log_2{((2^{x+5}-3)+3)}-5\\\\&amp;=\\log_2{2^{x+5}}-5\\\\&amp;=(x+5)\\log_2{2}-5\\\\&amp;=(x+5)\\times1-5\\\\&amp;=x+5-5\\\\&amp;=x\\end{aligned}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}f^{-1}(f(x))&amp;=2^{(\\log_2{(x+3)}-5)+5}-3\\\\&amp;=2^{(\\log_2{(x+3)}}-3\\\\&amp;=(x+3)-3\\\\&amp;=x\\end{aligned}[\/latex]<\/p>\r\nTherefore, the inverse function\u00a0[latex]f^{-1}(x)=2^{x+5}-3[\/latex] is correct because [latex]f(f^{-1}(x))=x[\/latex] and\u00a0[latex]f^{-1}(f(x))=x[\/latex].\r\n\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"textbox learning-objectives\">\n<h1>Learning Objectives<\/h1>\n<ul>\n<li>Apply the product rule<\/li>\n<li>Apply the quotient rule<\/li>\n<li>Apply the power rule<\/li>\n<li>Apply the change of base rule<\/li>\n<li>Explain natural logarithms and common logarithms<\/li>\n<\/ul>\n<\/div>\n<p>SInce logarithms are exponents, they have a list of rules that go along with them.<\/p>\n<h2>The Identity Property for Logarithms<\/h2>\n<p>The definition of logarithm states that [latex]\\log_b{a}=x[\/latex] is equivalent to [latex]a=b^x[\/latex]. Consequently, since\u00a0[latex]\\log_b{x}=\\log_b{x}[\/latex] then [latex]x=b^{\\log_b{x}}[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Identity property for Logarithms<\/h3>\n<p>For all real numbers [latex]b>0, b\\neq1[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]b^{\\log_b{x}}=x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Evaluate:<\/p>\n<ol>\n<li>[latex]2^{\\log_2{5}}[\/latex]<\/li>\n<li>[latex]3^{\\log_3{7}}[\/latex]<\/li>\n<li>[latex]7^{\\log_7{2}}[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>SInce\u00a0[latex]b^{\\log_b{x}}=x[\/latex],<\/p>\n<ol>\n<li>[latex]2^{\\log_2{5}}=5[\/latex]<\/li>\n<li>[latex]3^{\\log_3{7}}=7[\/latex]<\/li>\n<li>[latex]7^{\\log_7{2}}=2[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Evaluate:<\/p>\n<ol>\n<li>[latex]2^{\\log_2{3}}[\/latex]<\/li>\n<li>[latex]9^{\\log_9{7}}[\/latex]<\/li>\n<li>[latex]2^{\\log_2{5}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm150\">Show Answer<\/span><\/p>\n<div id=\"qhjm150\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>3<\/li>\n<li>7<\/li>\n<li>5<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>The Product Property for Logarithms<\/h2>\n<p>The product Property for logarithms mimics the product Property for exponents. SInce logarithms are exponents the exponential property [latex]a^m\\cdot a^n=a^{m+n}[\/latex] gets translated into logarithmic form. The multiplication of terms inside the argument of a logarithm is equal to the addition of logarithms of each term.<\/p>\n<div class=\"textbox shaded\">\n<h3>The Product Property for logarithms<\/h3>\n<p>For any positive real numbers [latex]b,\\;M,\\;N[\/latex], and [latex]b\\neq1[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\log_b{MN} = \\log_b{M}+\\log_b{N}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Expand the logarithm\u00a0[latex]\\log_3{3x}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>The argument consists of the product of 3 and [latex]x[\/latex], so applying the product property:<\/p>\n<p style=\"text-align: center;\">[latex]\\log_3{3x}=\\log_3{3}+\\log_3{x}[\/latex]<\/p>\n<p>We can then evaluate [latex]\\log_3{3}=1[\/latex], to get:<\/p>\n<p style=\"text-align: center;\">[latex]\\log_3{3}+\\log_3{x}=1+\\log_3x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Expand the logarithm\u00a0[latex]\\log_5{[5s(t-3)]}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>The argument consists of the product of [latex]5\\cdot s\\cdot (t-3)[\/latex], so we can expand the product property:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\log_5{[5s(t-3)]} &= \\log_5{5}+\\log_5{s} +\\log_5{(t-3)}\\\\&=1+\\log_5{s}+\\log_5{(t-3)}\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Expand the logarithm\u00a0[latex]\\log_4{16x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm311\">Show Answer<\/span><\/p>\n<div id=\"qhjm311\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]2+\\log_4x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Expand the logarithm\u00a0[latex]\\log_2{[8(s+2)(t-5)]}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm407\">Show Answer<\/span><\/p>\n<div id=\"qhjm407\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]3+\\log_2(s+2)+\\log_2(t-5)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>The Quotient Property for Logarithms<\/h2>\n<p>The quotient property for logarithms mimics the quotient property for exponents. SInce logarithms are exponents the exponential property [latex]\\dfrac{a^m}{a^n}=a^{m-n}[\/latex] gets translated into logarithmic form. The division of terms inside the argument of a logarithm is equal to the subtraction of logarithms of each term.<\/p>\n<div class=\"textbox shaded\">\n<h3>The quotient Property for logarithms<\/h3>\n<p style=\"text-align: left;\">For any positive real numbers [latex]b,\\;M,\\;N[\/latex], and [latex]b\\neq1[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\log_b{\\dfrac{M}{N}} = \\log_b{M}-\\log_b{N}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>Expand the logarithm\u00a0[latex]\\log_{10}{\\frac{x}{10}}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>The argument of the log consists of a quotient of the two terms\u00a0[latex]x[\/latex] and 10, so we apply the quotient property:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\log_{10}{\\frac{x}{10}} &= \\log_{10}{x}-\\log_{10}{10} \\\\&=\\log_{10}{x}-1\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>Expand the logarithm\u00a0[latex]\\log_{4}{\\frac{x}{4}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm492\">Show Answer<\/span><\/p>\n<div id=\"qhjm492\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\log_4x-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>The Power Property for Logarithms<\/h2>\n<p>The power Property comes from the product rule of exponents. [latex]\\log_b{M^r}=\\log_b{(M\\cdot M\\cdot M...M\\cdot M)}=\\log_bM+\\log_bM+\\log_bM+...+\\log_bM=r\\log_bM[\/latex]. If the argument of a logarithm is a single term with an exponent, we may pull the exponent in front of the logarithm.<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: left;\">The Power Property for Logarithms<\/h3>\n<p>For any positive real numbers [latex]b,\\;M,\\;N[\/latex], and [latex]b\\neq1[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\log_b{M^r}=r\\times\\log_b{M}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 5<\/h3>\n<p>Expand the logarithm\u00a0[latex]\\log_{2}{x^4}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>The argument of the logarithm is [latex]x^4[\/latex], a single term raised to a power, so we can apply the power property of logarithms:<\/p>\n<p style=\"text-align: center;\">[latex]\\log_{2}{x^4}=4\\log_2{x}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>Expand the logarithm\u00a0[latex]\\log_{7}{x^{-2}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm920\">Show Answer<\/span><\/p>\n<div id=\"qhjm920\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-2\\log_7x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Expanding a Logarithm<\/h2>\n<p>The properties we have learned so far to expand a logarithm can be used in conjunction with one another.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 6<\/h3>\n<p>Expand the logarithm [latex]\\log_{2}{\\frac{2y}{5}}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\log_{2}{\\frac{2y}{5}} &=\\log_2{(2y)}-\\log_2{5}&&\\text{Quotient Property}\\\\&=\\log_2{2}+\\log_2{y}-\\log_2{5}&&\\text{Product Property}\\\\&=1+\\log_2{y}-\\log_2{5}&&\\text{IdentityProperty}\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 7<\/h3>\n<p>Expand the logarithm [latex]\\log_{4}{\\frac{16y^3}{5}}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\log_{4}{\\frac{16y^3}{5}} &=\\log_4{(16y^3)}-\\log_4{5}&&\\text{Quotient Property}\\\\&=\\log_4{16}+\\log_4{y^3}-\\log_4{5}&&\\text{Product Property}\\\\&=\\log_4{4^2}+3\\log_4{y}-\\log_4{5}&&\\text{Power Property}\\\\&=2+3\\log_4{y}-\\log_4{5}&&\\text{Identity Property}\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 6<\/h3>\n<p>Expand the logarithm [latex]\\log_{10}{100x^3y^5}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm882\">Show Answer<\/span><\/p>\n<div id=\"qhjm882\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]2+3\\log_{10}x+5\\log_{10}y[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 7<\/h3>\n<p>Expand the logarithm [latex]\\log_{2}{\\dfrac{32x}{y^5}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm376\">Show Answer<\/span><\/p>\n<div id=\"qhjm376\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]5+\\log_2x-5\\log_2y[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Combining Logarithms<\/h2>\n<p>The properties for exponents we applied to expand a logarithm can be also be used (in reverse) to combine logarithms into a single logarithm. The assumption is that these logarithms have the same base.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 8<\/h3>\n<p>Combine [latex]\\log_4{2}+\\log_4{10}-log_4{5}[\/latex] into a single logarithm.<\/p>\n<h4>Solution<\/h4>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\log_4{2}+\\log_4{10}-\\log_4{5}&=\\log_4{(2\\times10)}-\\log_4{5}&&\\text{Product Property}\\\\&=\\log_4{\\frac{20}{5}}&&\\text{Quotient Property}\\\\&=\\log_4{4}&&\\text{Simplify}\\\\&=1\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 9<\/h3>\n<p>Combine [latex]2\\log_2{(x-2)}-\\log_2{(x^2-x-2)}[\/latex] into a single logarithm.<\/p>\n<h4>Solution<\/h4>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}2\\log_2{(x-2)}-\\log_2{(x^2-x-2)}&=\\log_2{(x-2)^2}-\\log_2{(x^2-x-2)}&&\\text{Power Property}\\\\&=\\log_2{\\frac{(x-2)^2}{x^2-x-2}}&&\\text{Quotient Property}\\\\&=\\log_2{\\frac{(x-2)^2}{(x-2)(x+1)}}&&\\text{Factor}\\\\&=\\log_2{\\frac{(x-2)}{(x+1)}}&&\\text{Simplify}\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 8<\/h3>\n<p>Combine [latex]\\log_2{3}+4\\log_2{x}+5\\log_2{y}[\/latex] into a single logarithm.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm202\">Show Answer<\/span><\/p>\n<div id=\"qhjm202\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\log_2{3x^4y^5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 9<\/h3>\n<p>Combine [latex]4\\log_2{x}-\\log_2{7}-5\\log_2{y}[\/latex] into a single logarithm.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm200\">Show Answer<\/span><\/p>\n<div id=\"qhjm200\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\log_2{\\dfrac{x^4}{7y^5}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 10<\/h3>\n<p>Combine [latex]2\\log_2{(x-1)}-\\log_2{(x^2-1)}[\/latex] into a single logarithm.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm201\">Show Answer<\/span><\/p>\n<div id=\"qhjm201\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\log_2{\\dfrac{x-1}{x+1}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Common and Natural Logarithms<\/h2>\n<p>If the base of a logarithm is 10, the logarithm is call a <em><strong>common logarithm<\/strong><\/em>. This is because the common number system we use in our daily life is base-10. If the base of a logarithm is the number [latex]e[\/latex] (also known as Euler&#8217;s number), the logarithm is call a <strong><em>natural logarithm<\/em><\/strong>. This is because the number [latex]e[\/latex] is an irrational number equal to 2.718281828459&#8230;; the ellipses &#8230; mean that the decimal never ends and never repeats. The number [latex]e[\/latex] was\u00a0first discovered in 1683 by the Swiss mathematician Jacob Bernoulli but was first evaluated by Leonhard Euler in 1737. Consequently, it is often referred to as Euler&#8217;s number.\u00a0The\u00a0natural\u00a0logarithm is one of the most useful functions in\u00a0mathematics, with applications\u00a0throughout the\u00a0physical\u00a0and biological sciences.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-3047\" style=\"orphans: 1;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/28010755\/log-and-ln-calaulator-195x300.png\" alt=\"Calculator buttons\" width=\"116\" height=\"179\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Most scientific calculators have both [latex]\\log_{10}[\/latex], which is written as [latex]\\log[\/latex] (the base is so <em>common<\/em> it is omitted) and [latex]\\log_e[\/latex] which is written [latex]\\ln[\/latex] ([latex]\\ln[\/latex] refers to\u00a0<em>logarithm naturale).<\/em><\/p>\n<p>Notice that behind the log button is [latex]10^x[\/latex], and behind the ln button is [latex]e^x[\/latex]. These are the inverse functions.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>Example 10<\/h3>\n<p>Use a calculator to evaluate the logarithms exactly or to 5 decimal places:<\/p>\n<ol>\n<li>[latex]\\log4[\/latex]<\/li>\n<li>[latex]\\log100[\/latex]<\/li>\n<li>[latex]\\ln4[\/latex]<\/li>\n<li>[latex]\\ln{e}[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>[latex]\\log4=0.60206[\/latex]<\/li>\n<li>[latex]\\log100=2[\/latex]<\/li>\n<li>[latex]\\ln{4}=1.38629[\/latex]<\/li>\n<li>[latex]\\ln{e}=1[\/latex]<\/li>\n<\/ol>\n<\/div>\n<p>All of the properties of logarithms apply to common and natural logarithms.<\/p>\n<div class=\"textbox shaded\">\n<h3>Properties of Logarithms<\/h3>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%;\">Property<\/th>\n<th style=\"width: 33.3333%;\">Base 10<\/th>\n<th style=\"width: 33.3333%;\">Base [latex]e[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">Identity<\/td>\n<td style=\"width: 33.3333%;\">[latex]\\log{10}=1[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]\\ln{e}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">Product<\/td>\n<td style=\"width: 33.3333%;\">[latex]\\log{MN}=\\log{M}+\\log{N}[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]\\ln{MN}=\\ln{M}+\\ln{N}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">Quotient<\/td>\n<td style=\"width: 33.3333%;\">[latex]\\log{\\dfrac{M}{N}}=\\log{M}-\\log{N}[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]\\ln{\\dfrac{M}{N}}=\\ln{M}-\\ln{N}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">Power<\/td>\n<td style=\"width: 33.3333%;\">[latex]\\log{M^r}=r\\log{M}[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]\\ln{M^r}=r\\ln{M}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 11<\/h3>\n<p>Evaluate [latex]\\ln{e^2}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\ln{e^2}&=2\\ln{e}\\\\&=2\\times1\\\\&=2\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 11<\/h3>\n<p>Evaluate [latex]\\log{10,000}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm869\">Show Answer<\/span><\/p>\n<div id=\"qhjm869\" class=\"hidden-answer\" style=\"display: none\">[latex]4[\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 12<\/h3>\n<p>Simplify to a single logarithm: [latex]\\log{(x^2+4x+3)}-2\\log{(x+1)}[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\log{(x^2+4x+3)}-2\\log{(x+1)}&=\\log{(x+3)(x+1)}-\\log{(x+1)^2}&&\\text{Factor. Power property.}\\\\&=\\log{\\dfrac{(x+3)(x+1)}{(x+1)^2}}&&\\text{Quotient property}\\\\&=\\log{\\dfrac{x+3}{x+1}}&&\\text{Simplify}\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 12<\/h3>\n<p>Simplify to a single logarithm:\u00a0[latex]4\\ln{x}-2\\ln{(x+1)}+\\ln{(x^2-1)}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm751\">Show Answer<\/span><\/p>\n<div id=\"qhjm751\" class=\"hidden-answer\" style=\"display: none\">[latex]\\ln{\\dfrac{x^4(x-1)}{(x+1)}}[\/latex]<\/div>\n<\/div>\n<\/div>\n<h2>The Change of Base Rule<\/h2>\n<p>Calculators are a great way to evaluate logarithms with base 10 or [latex]e[\/latex]. But what if we want to evaluate a logarithm with a different base? The good news is that we can change the base of any logarithm to any base we wish to use, most importantly base 10 and [latex]e[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>The Change of Base Formula<\/h3>\n<p>For any positive real number [latex]m[\/latex] with [latex]m\\neq1[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\log_b{a}=\\dfrac{\\log_m{a}}{\\log_m{b}}[\/latex]<\/p>\n<\/div>\n<p>SInce we usually have access to a calculator with logarithms of base 10 and [latex]e[\/latex], it is important to be able to change the base of any logarithm to 10 or\u00a0[latex]e[\/latex]. The change of base formula can be used:<\/p>\n<p style=\"text-align: center;\">[latex]\\log_ba=\\dfrac{\\log{a}}{\\log{b}}[\/latex]\u00a0 \u00a0 and\u00a0 \u00a0\u00a0[latex]\\log_ba=\\dfrac{\\ln{a}}{\\ln{b}}[\/latex]<\/p>\n<div class=\"textbox examples\">\n<h3>Example 13<\/h3>\n<p>Evaluate\u00a0[latex]\\log_2{5}[\/latex] to 5 decimal places.<\/p>\n<h4>Solution<\/h4>\n<p>Converting to common logs:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\log_2{5}&=\\dfrac{\\log{5}}{\\log{2}}\\\\&=2.32193\\end{aligned}[\/latex]<\/p>\n<p style=\"text-align: center;\">or<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\\log_2{5}&=\\dfrac{\\ln{5}}{\\ln{2}}\\\\&=2.32193\\end{aligned}[\/latex]<\/p>\n<p>Notice that when using a calculator we never write the answer to the numerator and denominator of a fraction, then divide those two rounded numbers. This is because rounding each number than dividing brings in rounding error. Instead we complete the full calculation in the calculator, then round at the very end.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 13<\/h3>\n<p>Evaluate [latex]\\log_9{32}[\/latex] to 5 decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm325\">Show Answer<\/span><\/p>\n<div id=\"qhjm325\" class=\"hidden-answer\" style=\"display: none\">[latex]1.57732[\/latex]<\/div>\n<\/div>\n<\/div>\n<h2>Composition with the Inverse Function (Application of the Properties of Logarithms)<\/h2>\n<p>In chapter 3.3.2, we learned that [latex]f(x)[\/latex] and [latex]g(x)[\/latex] are inverses of each other if\u00a0[latex]f\\left(g\\left(x\\right)\\right)=x[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)=x[\/latex]. We may use this property to determine if the inverse function we found is correct or not. For example, suppose someone found that the inverse function of [latex]f(x)=\\log_2{(x+3)}-5[\/latex] is [latex]f^{-1}(x)=2^{x+5}-3[\/latex]. We may justify this answer by checking if [latex]f(f^{-1}(x))=x[\/latex] and\u00a0[latex]f^{-1}(f(x))=x[\/latex]. If either one is false, then the inverse function is incorrect.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}f(f^{-1}(x))&=\\log_2{((2^{x+5}-3)+3)}-5\\\\&=\\log_2{2^{x+5}}-5\\\\&=(x+5)\\log_2{2}-5\\\\&=(x+5)\\times1-5\\\\&=x+5-5\\\\&=x\\end{aligned}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}f^{-1}(f(x))&=2^{(\\log_2{(x+3)}-5)+5}-3\\\\&=2^{(\\log_2{(x+3)}}-3\\\\&=(x+3)-3\\\\&=x\\end{aligned}[\/latex]<\/p>\n<p>Therefore, the inverse function\u00a0[latex]f^{-1}(x)=2^{x+5}-3[\/latex] is correct because [latex]f(f^{-1}(x))=x[\/latex] and\u00a0[latex]f^{-1}(f(x))=x[\/latex].<\/p>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\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-2511\">\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>Properties of Logarithm. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Examples and Try Its: hjm200; hjm201; hjm202; hjm325; hjm751; hjm869; hjm376; hjm882; hjm920; hjm492; hjm407; hjm311; hjm150. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <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":422608,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Properties of Logarithm\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Examples and Try Its: hjm200; hjm201; hjm202; hjm325; hjm751; hjm869; hjm376; hjm882; hjm920; hjm492; hjm407; hjm311; hjm150\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"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-2511","chapter","type-chapter","status-publish","hentry"],"part":2310,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2511","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":54,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2511\/revisions"}],"predecessor-version":[{"id":4708,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2511\/revisions\/4708"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/2310"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2511\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=2511"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2511"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=2511"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=2511"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}