{"id":11195,"date":"2015-07-14T18:46:41","date_gmt":"2015-07-14T18:46:41","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&#038;p=11195"},"modified":"2015-09-09T20:41:24","modified_gmt":"2015-09-09T20:41:24","slug":"key-terms-glossary-7","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/key-terms-glossary-7\/","title":{"raw":"Key Concepts","rendered":"Key Concepts"},"content":{"raw":"<section id=\"fs-id1165137749167\" class=\"key-equations\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Key Equations<\/h2>\r\n<table id=\"fs-id1737642\" summary=\"...\">\r\n<tbody>\r\n<tr>\r\n<td>General Form for the Translation of the Parent Logarithmic Function [latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\r\n<td>[latex] f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section id=\"fs-id1165137863125\" class=\"key-concepts\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Key Concepts<\/h2>\r\n<ul id=\"fs-id1165137863132\">\r\n\t<li>To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve for <em>x<\/em>.<\/li>\r\n\t<li>The graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] has an <em data-effect=\"italics\">x-<\/em>intercept at [latex]\\left(1,0\\right)[\/latex], domain [latex]\\left(0,\\infty \\right)[\/latex], range [latex]\\left(-\\infty ,\\infty \\right)[\/latex], vertical asymptote <em>x\u00a0<\/em>= 0, and\r\n<ul id=\"fs-id1165135441773\">\r\n\t<li>if <em>b\u00a0<\/em>&gt; 1, the function is increasing.<\/li>\r\n\t<li>if 0 &lt; <em>b\u00a0<\/em>&lt; 1, the function is decreasing.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex] shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] horizontally\r\n<ul id=\"fs-id1165135512562\">\r\n\t<li>left <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&gt; 0.<\/li>\r\n\t<li>right <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&lt; 0.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex] shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically\r\n<ul id=\"fs-id1165137761068\">\r\n\t<li>up <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&gt; 0.<\/li>\r\n\t<li>down <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&lt; 0.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>For any constant <em>a\u00a0<\/em>&gt; 0, the equation [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]\r\n<ul id=\"fs-id1165134040579\">\r\n\t<li>stretches the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of <em>a<\/em>\u00a0if |<em>a<\/em>| &gt; 1.<\/li>\r\n\t<li>compresses the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of <em>a<\/em>\u00a0if |<em>a<\/em>| &lt; 1.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>When the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by \u20131, the result is a reflection about the <em data-effect=\"italics\">x<\/em>-axis. When the input is multiplied by \u20131, the result is a reflection about the <em data-effect=\"italics\">y<\/em>-axis.\r\n<ul id=\"fs-id1165135186594\">\r\n\t<li>The equation [latex]f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex] represents a reflection of the parent function about the <em data-effect=\"italics\">x-<\/em>axis.<\/li>\r\n\t<li>The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex] represents a reflection of the parent function about the <em data-effect=\"italics\">y-<\/em>axis.<\/li>\r\n<\/ul>\r\n<ul id=\"fs-id1165137834414\">\r\n\t<li>A graphing calculator may be used to approximate solutions to some logarithmic equations.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>All translations of the logarithmic function can be summarized by the general equation [latex] f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex].<\/li>\r\n\t<li>Given an equation with the general form [latex] f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex], we can identify the vertical asymptote <em>x\u00a0<\/em>= \u2013c for the transformation.<\/li>\r\n\t<li>Using the general equation [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex], we can write the equation of a logarithmic function given its graph.<\/li>\r\n<\/ul>\r\n<\/section>","rendered":"<section id=\"fs-id1165137749167\" class=\"key-equations\" data-depth=\"1\">\n<h2 data-type=\"title\">Key Equations<\/h2>\n<table id=\"fs-id1737642\" summary=\"...\">\n<tbody>\n<tr>\n<td>General Form for the Translation of the Parent Logarithmic Function [latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165137863125\" class=\"key-concepts\" data-depth=\"1\">\n<h2 data-type=\"title\">Key Concepts<\/h2>\n<ul id=\"fs-id1165137863132\">\n<li>To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve for <em>x<\/em>.<\/li>\n<li>The graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] has an <em data-effect=\"italics\">x-<\/em>intercept at [latex]\\left(1,0\\right)[\/latex], domain [latex]\\left(0,\\infty \\right)[\/latex], range [latex]\\left(-\\infty ,\\infty \\right)[\/latex], vertical asymptote <em>x\u00a0<\/em>= 0, and\n<ul id=\"fs-id1165135441773\">\n<li>if <em>b\u00a0<\/em>&gt; 1, the function is increasing.<\/li>\n<li>if 0 &lt; <em>b\u00a0<\/em>&lt; 1, the function is decreasing.<\/li>\n<\/ul>\n<\/li>\n<li>The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex] shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] horizontally\n<ul id=\"fs-id1165135512562\">\n<li>left <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&gt; 0.<\/li>\n<li>right <em>c<\/em>\u00a0units if <em>c\u00a0<\/em>&lt; 0.<\/li>\n<\/ul>\n<\/li>\n<li>The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex] shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically\n<ul id=\"fs-id1165137761068\">\n<li>up <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&gt; 0.<\/li>\n<li>down <em>d<\/em>\u00a0units if <em>d\u00a0<\/em>&lt; 0.<\/li>\n<\/ul>\n<\/li>\n<li>For any constant <em>a\u00a0<\/em>&gt; 0, the equation [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]\n<ul id=\"fs-id1165134040579\">\n<li>stretches the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of <em>a<\/em>\u00a0if |<em>a<\/em>| &gt; 1.<\/li>\n<li>compresses the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of <em>a<\/em>\u00a0if |<em>a<\/em>| &lt; 1.<\/li>\n<\/ul>\n<\/li>\n<li>When the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by \u20131, the result is a reflection about the <em data-effect=\"italics\">x<\/em>-axis. When the input is multiplied by \u20131, the result is a reflection about the <em data-effect=\"italics\">y<\/em>-axis.\n<ul id=\"fs-id1165135186594\">\n<li>The equation [latex]f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex] represents a reflection of the parent function about the <em data-effect=\"italics\">x-<\/em>axis.<\/li>\n<li>The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex] represents a reflection of the parent function about the <em data-effect=\"italics\">y-<\/em>axis.<\/li>\n<\/ul>\n<ul id=\"fs-id1165137834414\">\n<li>A graphing calculator may be used to approximate solutions to some logarithmic equations.<\/li>\n<\/ul>\n<\/li>\n<li>All translations of the logarithmic function can be summarized by the general equation [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex].<\/li>\n<li>Given an equation with the general form [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex], we can identify the vertical asymptote <em>x\u00a0<\/em>= \u2013c for the transformation.<\/li>\n<li>Using the general equation [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex], we can write the equation of a logarithmic function given its graph.<\/li>\n<\/ul>\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-11195\">\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":4,"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-11195","chapter","type-chapter","status-publish","hentry"],"part":11188,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/chapters\/11195","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/chapters\/11195\/revisions"}],"predecessor-version":[{"id":12974,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/chapters\/11195\/revisions\/12974"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/parts\/11188"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/chapters\/11195\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/wp\/v2\/media?parent=11195"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/chapter-type?post=11195"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/wp\/v2\/contributor?post=11195"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/wp\/v2\/license?post=11195"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}