{"id":2082,"date":"2016-11-03T17:18:15","date_gmt":"2016-11-03T17:18:15","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=2082"},"modified":"2025-10-15T21:47:22","modified_gmt":"2025-10-15T21:47:22","slug":"summary-logarithmic-properties","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/chapter\/summary-logarithmic-properties\/","title":{"raw":"Summary: Properties of Logarithms","rendered":"Summary: Properties of Logarithms"},"content":{"raw":"<h2>Key Equations<\/h2>\r\n<table summary=\"...\">\r\n<tbody>\r\n<tr>\r\n<td>The Product Rule for Logarithms<\/td>\r\n<td>[latex]{\\mathrm{log}}_{b}\\left(MN\\right)={\\mathrm{log}}_{b}\\left(M\\right)+{\\mathrm{log}}_{b}\\left(N\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The Quotient Rule for Logarithms<\/td>\r\n<td>[latex]{\\mathrm{log}}_{b}\\left(\\frac{M}{N}\\right)={\\mathrm{log}}_{b}M-{\\mathrm{log}}_{b}N[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The Power Rule for Logarithms<\/td>\r\n<td>[latex]{\\mathrm{log}}_{b}\\left({M}^{n}\\right)=n{\\mathrm{log}}_{b}M[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The Change-of-Base Formula<\/td>\r\n<td>[latex]{\\mathrm{log}}_{b}M\\text{=}\\frac{{\\mathrm{log}}_{n}M}{{\\mathrm{log}}_{n}b}\\text{ }n&gt;0,n\\ne 1,b\\ne 1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms.<\/li>\r\n \t<li>We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms.<\/li>\r\n \t<li>We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base.<\/li>\r\n \t<li>We can use the product rule, quotient rule, and power rule together to combine or expand a logarithm with a complex input.<\/li>\r\n \t<li>The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm.<\/li>\r\n \t<li>We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base formula.<\/li>\r\n \t<li>The change-of-base formula is often used to rewrite a logarithm with a base other than 10 or [latex]e[\/latex]\u00a0as the quotient of natural or common logs. A calculator can then be used to evaluate it.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165135397912\" class=\"definition\">\r\n \t<dt><strong>change-of-base formula<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135397918\">a formula for converting a logarithm with any base to a quotient of logarithms with any other base<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135397926\" class=\"definition\">\r\n \t<dt><strong>power rule for logarithms<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135397932\">a rule of logarithms that states that the log of a power is equal to the product of the exponent and the log of its base<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137838635\" class=\"definition\">\r\n \t<dt><strong>product rule for logarithms<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137838640\">a rule of logarithms that states that the log of a product is equal to a sum of logarithms<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137838635\" class=\"definition\">\r\n \t<dt>\r\n<dl id=\"fs-id1165137838635\" class=\"definition\">\r\n \t<dt><strong>quotient rule for logarithms<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137838640\">a rule of logarithms that states that the log of a quotient is equal to a difference of logarithms<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n<\/dl>\r\n","rendered":"<h2>Key Equations<\/h2>\n<table summary=\"...\">\n<tbody>\n<tr>\n<td>The Product Rule for Logarithms<\/td>\n<td>[latex]{\\mathrm{log}}_{b}\\left(MN\\right)={\\mathrm{log}}_{b}\\left(M\\right)+{\\mathrm{log}}_{b}\\left(N\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The Quotient Rule for Logarithms<\/td>\n<td>[latex]{\\mathrm{log}}_{b}\\left(\\frac{M}{N}\\right)={\\mathrm{log}}_{b}M-{\\mathrm{log}}_{b}N[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The Power Rule for Logarithms<\/td>\n<td>[latex]{\\mathrm{log}}_{b}\\left({M}^{n}\\right)=n{\\mathrm{log}}_{b}M[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The Change-of-Base Formula<\/td>\n<td>[latex]{\\mathrm{log}}_{b}M\\text{=}\\frac{{\\mathrm{log}}_{n}M}{{\\mathrm{log}}_{n}b}\\text{ }n>0,n\\ne 1,b\\ne 1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms.<\/li>\n<li>We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms.<\/li>\n<li>We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base.<\/li>\n<li>We can use the product rule, quotient rule, and power rule together to combine or expand a logarithm with a complex input.<\/li>\n<li>The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm.<\/li>\n<li>We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base formula.<\/li>\n<li>The change-of-base formula is often used to rewrite a logarithm with a base other than 10 or [latex]e[\/latex]\u00a0as the quotient of natural or common logs. A calculator can then be used to evaluate it.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135397912\" class=\"definition\">\n<dt><strong>change-of-base formula<\/strong><\/dt>\n<dd id=\"fs-id1165135397918\">a formula for converting a logarithm with any base to a quotient of logarithms with any other base<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135397926\" class=\"definition\">\n<dt><strong>power rule for logarithms<\/strong><\/dt>\n<dd id=\"fs-id1165135397932\">a rule of logarithms that states that the log of a power is equal to the product of the exponent and the log of its base<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137838635\" class=\"definition\">\n<dt><strong>product rule for logarithms<\/strong><\/dt>\n<dd id=\"fs-id1165137838640\">a rule of logarithms that states that the log of a product is equal to a sum of logarithms<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137838635\" class=\"definition\">\n<dt>\n<\/dt>\n<dt><strong>quotient rule for logarithms<\/strong><\/dt>\n<dd id=\"fs-id1165137838640\">a rule of logarithms that states that the log of a quotient is equal to a difference of logarithms<\/dd>\n<\/dl>\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-2082\">\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>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/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\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/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":21,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et 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