{"id":168,"date":"2021-02-03T22:26:42","date_gmt":"2021-02-03T22:26:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=168"},"modified":"2021-03-15T20:27:42","modified_gmt":"2021-03-15T20:27:42","slug":"summary-of-exponential-and-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/summary-of-exponential-and-logarithmic-functions\/","title":{"raw":"Summary of Exponential and Logarithmic Functions","rendered":"Summary of Exponential and Logarithmic Functions"},"content":{"raw":"<div id=\"fs-id1170572176246\" class=\"learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1170572216270\">\r\n \t<li>The exponential function [latex]y=b^x[\/latex] is increasing if [latex]b&gt;1[\/latex] and decreasing if [latex]0&lt;b&lt;1[\/latex]. Its domain is [latex](\u2212\\infty ,\\infty)[\/latex] and its range is [latex](0,\\infty)[\/latex].<\/li>\r\n \t<li>The logarithmic function [latex]y=\\log_b(x)[\/latex] is the inverse of [latex]y=b^x[\/latex]. Its domain is [latex](0,\\infty)[\/latex] and its range is [latex](\u2212\\infty,\\infty)[\/latex].<\/li>\r\n \t<li>The natural exponential function is [latex]y=e^x[\/latex] and the natural logarithmic function is [latex]y=\\ln x=\\log_e x[\/latex].<\/li>\r\n \t<li>Given an exponential function or logarithmic function in base [latex]a[\/latex], we can make a change of base to convert this function to any base [latex]b&gt;0, \\, b \\ne 1[\/latex]. We typically convert to base [latex]e[\/latex].<\/li>\r\n \t<li>The hyperbolic functions involve combinations of the exponential functions [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]. As a result, the inverse hyperbolic functions involve the natural logarithm.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170572544592\" class=\"definition\">\r\n \t<dt>base<\/dt>\r\n \t<dd id=\"fs-id1170572544597\">the number [latex]b[\/latex] in the exponential function [latex]f(x)=b^x[\/latex] and the logarithmic function [latex]f(x)=\\log_b x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n \t<dt>exponent<\/dt>\r\n \t<dd id=\"fs-id1170572544654\">the value [latex]x[\/latex] in the expression [latex]b^x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544670\" class=\"definition\">\r\n \t<dt>hyperbolic functions<\/dt>\r\n \t<dd id=\"fs-id1170572544676\">the functions denoted [latex]\\sinh, \\, \\cosh, \\, \\tanh, \\, \\text{csch}, \\, \\text{sech}[\/latex], and [latex]\\coth[\/latex], which involve certain combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572294452\" class=\"definition\">\r\n \t<dt>inverse hyperbolic functions<\/dt>\r\n \t<dd id=\"fs-id1170572294458\">the inverses of the hyperbolic functions where [latex]\\cosh[\/latex] and [latex]\\text{sech}[\/latex] are restricted to the domain [latex][0,\\infty)[\/latex]; each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572294493\" class=\"definition\">\r\n \t<dt>natural exponential function<\/dt>\r\n \t<dd id=\"fs-id1170572294499\">the function [latex]f(x)=e^x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572294522\" class=\"definition\">\r\n \t<dt>natural logarithm<\/dt>\r\n \t<dd id=\"fs-id1170572294527\">the function [latex]\\ln x=\\log_e x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572294549\" class=\"definition\">\r\n \t<dt>number e<\/dt>\r\n \t<dd id=\"fs-id1170572294554\">as [latex]m[\/latex] gets larger, the quantity [latex](1+(1\/m))^m[\/latex] gets closer to some real number; we define that real number to be [latex]e[\/latex]; the value of [latex]e[\/latex] is approximately 2.718282<\/dd>\r\n<\/dl>","rendered":"<div id=\"fs-id1170572176246\" class=\"learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1170572216270\">\n<li>The exponential function [latex]y=b^x[\/latex] is increasing if [latex]b>1[\/latex] and decreasing if [latex]0<b<1[\/latex]. Its domain is [latex](\u2212\\infty ,\\infty)[\/latex] and its range is [latex](0,\\infty)[\/latex].<\/li>\n<li>The logarithmic function [latex]y=\\log_b(x)[\/latex] is the inverse of [latex]y=b^x[\/latex]. Its domain is [latex](0,\\infty)[\/latex] and its range is [latex](\u2212\\infty,\\infty)[\/latex].<\/li>\n<li>The natural exponential function is [latex]y=e^x[\/latex] and the natural logarithmic function is [latex]y=\\ln x=\\log_e x[\/latex].<\/li>\n<li>Given an exponential function or logarithmic function in base [latex]a[\/latex], we can make a change of base to convert this function to any base [latex]b>0, \\, b \\ne 1[\/latex]. We typically convert to base [latex]e[\/latex].<\/li>\n<li>The hyperbolic functions involve combinations of the exponential functions [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]. As a result, the inverse hyperbolic functions involve the natural logarithm.<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170572544592\" class=\"definition\">\n<dt>base<\/dt>\n<dd id=\"fs-id1170572544597\">the number [latex]b[\/latex] in the exponential function [latex]f(x)=b^x[\/latex] and the logarithmic function [latex]f(x)=\\log_b x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572544649\" class=\"definition\">\n<dt>exponent<\/dt>\n<dd id=\"fs-id1170572544654\">the value [latex]x[\/latex] in the expression [latex]b^x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572544670\" class=\"definition\">\n<dt>hyperbolic functions<\/dt>\n<dd id=\"fs-id1170572544676\">the functions denoted [latex]\\sinh, \\, \\cosh, \\, \\tanh, \\, \\text{csch}, \\, \\text{sech}[\/latex], and [latex]\\coth[\/latex], which involve certain combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572294452\" class=\"definition\">\n<dt>inverse hyperbolic functions<\/dt>\n<dd id=\"fs-id1170572294458\">the inverses of the hyperbolic functions where [latex]\\cosh[\/latex] and [latex]\\text{sech}[\/latex] are restricted to the domain [latex][0,\\infty)[\/latex]; each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572294493\" class=\"definition\">\n<dt>natural exponential function<\/dt>\n<dd id=\"fs-id1170572294499\">the function [latex]f(x)=e^x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572294522\" class=\"definition\">\n<dt>natural logarithm<\/dt>\n<dd id=\"fs-id1170572294527\">the function [latex]\\ln x=\\log_e x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572294549\" class=\"definition\">\n<dt>number e<\/dt>\n<dd id=\"fs-id1170572294554\">as [latex]m[\/latex] gets larger, the quantity [latex](1+(1\/m))^m[\/latex] gets closer to some real number; we define that real number to be [latex]e[\/latex]; the value of [latex]e[\/latex] is approximately 2.718282<\/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-168\">\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>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/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":17533,"menu_order":25,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-168","chapter","type-chapter","status-publish","hentry"],"part":21,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/168","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/168\/revisions"}],"predecessor-version":[{"id":1256,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/168\/revisions\/1256"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/21"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/168\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=168"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=168"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=168"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=168"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}