{"id":1563,"date":"2015-11-12T18:35:28","date_gmt":"2015-11-12T18:35:28","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1563"},"modified":"2015-11-12T18:35:28","modified_gmt":"2015-11-12T18:35:28","slug":"key-concepts-glossary-36","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/chapter\/key-concepts-glossary-36\/","title":{"raw":"Key Concepts &amp; Glossary","rendered":"Key Concepts &amp; Glossary"},"content":{"raw":"<section id=\"fs-id1165137870892\" class=\"key-equations\" data-depth=\"1\"><h1 data-type=\"title\">Key Equations<\/h1>\n<table id=\"fs-id1983134\" summary=\"...\"><tbody><tr><td>Definition of the logarithmic function<\/td>\n<td>For [latex]\\text{ } x&gt;0,b&gt;0,b\\ne 1\\\\[\/latex],[latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] if and only if [latex]\\text{ }{b}^{y}=x\\\\[\/latex].<\/td>\n<\/tr><tr><td>Definition of the common logarithm<\/td>\n<td>For [latex]\\text{ }x&gt;0\\\\[\/latex], [latex]y=\\mathrm{log}\\left(x\\right)\\\\[\/latex] if and only if [latex]\\text{ }{10}^{y}=x\\\\[\/latex].<\/td>\n<\/tr><tr><td>Definition of the natural logarithm<\/td>\n<td>For [latex]\\text{ }x&gt;0\\\\[\/latex], [latex]y=\\mathrm{ln}\\left(x\\right)\\\\[\/latex] if and only if [latex]\\text{ }{e}^{y}=x\\\\[\/latex].<\/td>\n<\/tr><\/tbody><\/table><\/section><section id=\"fs-id1165135699130\" class=\"key-concepts\" data-depth=\"1\"><h1 data-type=\"title\">Key Concepts<\/h1>\n<ul id=\"fs-id1165137574258\"><li>The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.<\/li>\n\t<li>Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm.<\/li>\n\t<li>Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm.<\/li>\n\t<li>Logarithmic functions with base <em>b<\/em>\u00a0can be evaluated mentally using previous knowledge of powers of <em>b<\/em>.<\/li>\n\t<li>Common logarithms can be evaluated mentally using previous knowledge of powers of 10.<\/li>\n\t<li>When common logarithms cannot be evaluated mentally, a calculator can be used.<\/li>\n\t<li>Real-world exponential problems with base 10\u00a0can be rewritten as a common logarithm and then evaluated using a calculator.<\/li>\n\t<li>Natural logarithms can be evaluated using a calculator.<\/li>\n<\/ul><h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl id=\"fs-id1165135160066\" class=\"definition\"><dt><strong>common logarithm<\/strong><\/dt><dd id=\"fs-id1165137571387\">the exponent to which 10 must be raised to get <em>x<\/em>; [latex]{\\mathrm{log}}_{10}\\left(x\\right)\\\\[\/latex] is written simply as [latex]\\mathrm{log}\\left(x\\right)\\\\[\/latex].<\/dd><\/dl><dl id=\"fs-id1165137780762\" class=\"definition\"><dt><strong>logarithm<\/strong><\/dt><dd id=\"fs-id1165137849198\">the exponent to which <em>b<\/em>\u00a0must be raised to get <em>x<\/em>; written [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex]<\/dd><\/dl><dl class=\"definition\"><dt><strong>natural logarithm<\/strong><\/dt><dd>the exponent to which the number <em>e<\/em>\u00a0must be raised to get <em>x<\/em>; [latex]{\\mathrm{log}}_{e}\\left(x\\right)\\\\[\/latex] is written as [latex]\\mathrm{ln}\\left(x\\right)\\\\[\/latex].<\/dd><\/dl><\/section>","rendered":"<section id=\"fs-id1165137870892\" class=\"key-equations\" data-depth=\"1\">\n<h1 data-type=\"title\">Key Equations<\/h1>\n<table id=\"fs-id1983134\" summary=\"...\">\n<tbody>\n<tr>\n<td>Definition of the logarithmic function<\/td>\n<td>For [latex]\\text{ } x>0,b>0,b\\ne 1\\\\[\/latex],[latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex] if and only if [latex]\\text{ }{b}^{y}=x\\\\[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Definition of the common logarithm<\/td>\n<td>For [latex]\\text{ }x>0\\\\[\/latex], [latex]y=\\mathrm{log}\\left(x\\right)\\\\[\/latex] if and only if [latex]\\text{ }{10}^{y}=x\\\\[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Definition of the natural logarithm<\/td>\n<td>For [latex]\\text{ }x>0\\\\[\/latex], [latex]y=\\mathrm{ln}\\left(x\\right)\\\\[\/latex] if and only if [latex]\\text{ }{e}^{y}=x\\\\[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165135699130\" class=\"key-concepts\" data-depth=\"1\">\n<h1 data-type=\"title\">Key Concepts<\/h1>\n<ul id=\"fs-id1165137574258\">\n<li>The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.<\/li>\n<li>Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm.<\/li>\n<li>Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm.<\/li>\n<li>Logarithmic functions with base <em>b<\/em>\u00a0can be evaluated mentally using previous knowledge of powers of <em>b<\/em>.<\/li>\n<li>Common logarithms can be evaluated mentally using previous knowledge of powers of 10.<\/li>\n<li>When common logarithms cannot be evaluated mentally, a calculator can be used.<\/li>\n<li>Real-world exponential problems with base 10\u00a0can be rewritten as a common logarithm and then evaluated using a calculator.<\/li>\n<li>Natural logarithms can be evaluated using a calculator.<\/li>\n<\/ul>\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl id=\"fs-id1165135160066\" class=\"definition\">\n<dt><strong>common logarithm<\/strong><\/dt>\n<dd id=\"fs-id1165137571387\">the exponent to which 10 must be raised to get <em>x<\/em>; [latex]{\\mathrm{log}}_{10}\\left(x\\right)\\\\[\/latex] is written simply as [latex]\\mathrm{log}\\left(x\\right)\\\\[\/latex].<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137780762\" class=\"definition\">\n<dt><strong>logarithm<\/strong><\/dt>\n<dd id=\"fs-id1165137849198\">the exponent to which <em>b<\/em>\u00a0must be raised to get <em>x<\/em>; written [latex]y={\\mathrm{log}}_{b}\\left(x\\right)\\\\[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt><strong>natural logarithm<\/strong><\/dt>\n<dd>the exponent to which the number <em>e<\/em>\u00a0must be raised to get <em>x<\/em>; [latex]{\\mathrm{log}}_{e}\\left(x\\right)\\\\[\/latex] is written as [latex]\\mathrm{ln}\\left(x\\right)\\\\[\/latex].<\/dd>\n<\/dl>\n<\/section>\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-1563\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><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><\/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":276,"menu_order":7,"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.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1563","chapter","type-chapter","status-publish","hentry"],"part":1552,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1563","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1563\/revisions"}],"predecessor-version":[{"id":2338,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1563\/revisions\/2338"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1552"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1563\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/media?parent=1563"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1563"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1563"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/wp-json\/wp\/v2\/license?post=1563"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}