{"id":111,"date":"2025-02-13T22:43:54","date_gmt":"2025-02-13T22:43:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/logarithmic-properties-2\/"},"modified":"2026-03-17T23:14:00","modified_gmt":"2026-03-17T23:14:00","slug":"logarithmic-properties-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/logarithmic-properties-2\/","title":{"raw":"Logarithmic Properties: Learn It 1","rendered":"Logarithmic Properties: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Expand logarithmic expressions.<\/li>\r\n \t<li>Condense logarithmic expressions.<\/li>\r\n \t<li>Use the change-of-base formula for logarithms.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165137657409\"><\/section>\r\n<div id=\"Example_04_05_09\" class=\"example\">\r\n<div id=\"fs-id1165137833837\" class=\"exercise\"><\/div>\r\n<\/div>\r\n<h2>Basic Properties of Logarithms<\/h2>\r\nMathematical properties and rules are essential tools for manipulating equations and expressions. While you may be familiar with properties of real numbers and exponents, logarithms have their own set of important properties. These logarithmic properties are fundamental for simplifying, expanding, and solving logarithmic expressions and equations. The zero property, identity property, and inverse property of logarithms form the foundation for understanding more complex logarithmic operations.\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">We can express the relationship between logarithmic form and its corresponding exponential form as follows:\r\n<p style=\"text-align: center;\">[latex]\\log_b\\left(x\\right)=y\\ \\Leftrightarrow \\ {b}^{y}=x,\\ \\text{}b&gt;0,\\ b\\ne 1[\/latex]<\/p>\r\nThat is, to say that the\u00a0<em>logarithm to base <\/em>[latex]b[\/latex]<em> of <\/em>\u00a0[latex]x[\/latex]<em> is\u00a0<\/em>[latex]y[\/latex]\u00a0is equivalent to saying that\u00a0[latex]y[\/latex]\u00a0<em>is the exponent on the base <\/em>[latex]b[\/latex]<em> that produces\u00a0<\/em>[latex]x[\/latex].\r\n\r\nNote that the base [latex]b[\/latex]\u00a0is always a positive number other than [latex]1[\/latex]\u00a0and\u00a0that the logarithmic and exponential forms \"undo\" each other.\r\n\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>zero property of logarithms<\/h3>\r\n<p style=\"text-align: center;\">[latex]\\log_b(1)=0[\/latex]<\/p>\r\n&nbsp;\r\n\r\nThis means that the logarithm of [latex]1[\/latex] to any base [latex]b[\/latex] (where [latex]b \\gt 0[\/latex] and [latex]b \\ne 1[\/latex]) is always [latex]0[\/latex].\r\n\r\n&nbsp;\r\n\r\nThe Zero Property of Logarithms, [latex]\\log_b(1)=0[\/latex], holds because we can rewrite it as [latex]b^0 = 1[\/latex], showing that any base [latex]b[\/latex] raised to the power of [latex]0[\/latex] equals [latex]1[\/latex].\r\n\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>identity property of logarithms<\/h3>\r\n<p style=\"text-align: center;\">[latex]\\log_b(b)=1[\/latex]<\/p>\r\n&nbsp;\r\n\r\nThe Identity Property of Logarithms, [latex]\\log_b(b)=1[\/latex], holds because, any base [latex]b[\/latex] raised to the power of [latex]1[\/latex] equals itself, i.e., [latex]b^1 = b[\/latex].\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p style=\"text-align: left;\">Use the the fact that exponentials and logarithms are inverses to prove the zero and identity exponent rule for\u00a0the following:<\/p>\r\n<p style=\"text-align: left;\">1. [latex]{\\mathrm{log}}_{5}1=0[\/latex]<\/p>\r\n<p style=\"text-align: left;\">2. [latex]{\\mathrm{log}}_{5}5=1[\/latex]\r\n[reveal-answer q=\"622755\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"622755\"]<\/p>\r\n<p style=\"text-align: left;\">1.[latex]{\\mathrm{log}}_{5}1=0[\/latex] \u00a0since [latex]{5}^{0}=1[\/latex]<\/p>\r\n<p style=\"text-align: left;\">2.[latex]{\\mathrm{log}}_{5}5=1[\/latex] since [latex]{5}^{1}=5[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321453[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321454[\/ohm_question]<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>inverse property of logarithms<\/h3>\r\nThe inverse property of logarithms involves both logarithms and exponents, showing how they undo each other.\r\n\r\n&nbsp;\r\n\r\nThe properties are:\r\n<ul>\r\n \t<li>[latex]b^{\\log_b(x)}=x[\/latex]<\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 40px;\">This means that if you take the logarithm of a number and then use that result as the exponent for the base of the logarithm, you get the original number.<\/p>\r\n\r\n<ul>\r\n \t<li>[latex]\\log_b(b^x)=x[\/latex]<\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 40px;\">This means that if you have a base raised to a power and then take the logarithm of that result, you get the exponent.<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Evaluate:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\mathrm{log}\\left(100\\right)[\/latex]<\/li>\r\n \t<li>[latex]{e}^{\\mathrm{ln}\\left(7\\right)}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"804776\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"804776\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>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].<\/li>\r\n \t<li>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]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321455[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321456[\/ohm_question]<\/section><section id=\"fs-id1165134049414\" class=\"key-concepts\">\r\n<dl id=\"fs-id1165137890674\" class=\"definition\">\r\n \t<dd id=\"fs-id1165137890679\"><\/dd>\r\n<\/dl>\r\n<\/section>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Expand logarithmic expressions.<\/li>\n<li>Condense logarithmic expressions.<\/li>\n<li>Use the change-of-base formula for logarithms.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<section id=\"fs-id1165137657409\"><\/section>\n<div id=\"Example_04_05_09\" class=\"example\">\n<div id=\"fs-id1165137833837\" class=\"exercise\"><\/div>\n<\/div>\n<h2>Basic Properties of Logarithms<\/h2>\n<p>Mathematical properties and rules are essential tools for manipulating equations and expressions. While you may be familiar with properties of real numbers and exponents, logarithms have their own set of important properties. These logarithmic properties are fundamental for simplifying, expanding, and solving logarithmic expressions and equations. The zero property, identity property, and inverse property of logarithms form the foundation for understanding more complex logarithmic operations.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">We can express the relationship between logarithmic form and its corresponding exponential form as follows:<\/p>\n<p style=\"text-align: center;\">[latex]\\log_b\\left(x\\right)=y\\ \\Leftrightarrow \\ {b}^{y}=x,\\ \\text{}b>0,\\ b\\ne 1[\/latex]<\/p>\n<p>That is, to say that the\u00a0<em>logarithm to base <\/em>[latex]b[\/latex]<em> of <\/em>\u00a0[latex]x[\/latex]<em> is\u00a0<\/em>[latex]y[\/latex]\u00a0is equivalent to saying that\u00a0[latex]y[\/latex]\u00a0<em>is the exponent on the base <\/em>[latex]b[\/latex]<em> that produces\u00a0<\/em>[latex]x[\/latex].<\/p>\n<p>Note that the base [latex]b[\/latex]\u00a0is always a positive number other than [latex]1[\/latex]\u00a0and\u00a0that the logarithmic and exponential forms &#8220;undo&#8221; each other.<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>zero property of logarithms<\/h3>\n<p style=\"text-align: center;\">[latex]\\log_b(1)=0[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>This means that the logarithm of [latex]1[\/latex] to any base [latex]b[\/latex] (where [latex]b \\gt 0[\/latex] and [latex]b \\ne 1[\/latex]) is always [latex]0[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>The Zero Property of Logarithms, [latex]\\log_b(1)=0[\/latex], holds because we can rewrite it as [latex]b^0 = 1[\/latex], showing that any base [latex]b[\/latex] raised to the power of [latex]0[\/latex] equals [latex]1[\/latex].<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>identity property of logarithms<\/h3>\n<p style=\"text-align: center;\">[latex]\\log_b(b)=1[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The Identity Property of Logarithms, [latex]\\log_b(b)=1[\/latex], holds because, any base [latex]b[\/latex] raised to the power of [latex]1[\/latex] equals itself, i.e., [latex]b^1 = b[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p style=\"text-align: left;\">Use the the fact that exponentials and logarithms are inverses to prove the zero and identity exponent rule for\u00a0the following:<\/p>\n<p style=\"text-align: left;\">1. [latex]{\\mathrm{log}}_{5}1=0[\/latex]<\/p>\n<p style=\"text-align: left;\">2. [latex]{\\mathrm{log}}_{5}5=1[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q622755\">Show Solution<\/button><\/p>\n<div id=\"q622755\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">1.[latex]{\\mathrm{log}}_{5}1=0[\/latex] \u00a0since [latex]{5}^{0}=1[\/latex]<\/p>\n<p style=\"text-align: left;\">2.[latex]{\\mathrm{log}}_{5}5=1[\/latex] since [latex]{5}^{1}=5[\/latex]<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321453\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321453&theme=lumen&iframe_resize_id=ohm321453&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321454\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321454&theme=lumen&iframe_resize_id=ohm321454&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>inverse property of logarithms<\/h3>\n<p>The inverse property of logarithms involves both logarithms and exponents, showing how they undo each other.<\/p>\n<p>&nbsp;<\/p>\n<p>The properties are:<\/p>\n<ul>\n<li>[latex]b^{\\log_b(x)}=x[\/latex]<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">This means that if you take the logarithm of a number and then use that result as the exponent for the base of the logarithm, you get the original number.<\/p>\n<ul>\n<li>[latex]\\log_b(b^x)=x[\/latex]<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">This means that if you have a base raised to a power and then take the logarithm of that result, you get the exponent.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\mathrm{log}\\left(100\\right)[\/latex]<\/li>\n<li>[latex]{e}^{\\mathrm{ln}\\left(7\\right)}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q804776\">Show Solution<\/button><\/p>\n<div id=\"q804776\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>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].<\/li>\n<li>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]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" 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