{"id":1189,"date":"2021-06-30T17:02:07","date_gmt":"2021-06-30T17:02:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-integrals-exponential-functions-and-logarithms\/"},"modified":"2021-11-17T02:12:25","modified_gmt":"2021-11-17T02:12:25","slug":"summary-of-integrals-exponential-functions-and-logarithms","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-integrals-exponential-functions-and-logarithms\/","title":{"raw":"Summary of Integrals, Exponential Functions, and Logarithms","rendered":"Summary of Integrals, Exponential Functions, and Logarithms"},"content":{"raw":"<div id=\"fs-id1167793423569\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1167793931233\">\r\n \t<li>The earlier treatment of logarithms and exponential functions did not define the functions precisely and formally. This section develops the concepts in a mathematically rigorous way.<\/li>\r\n \t<li>The cornerstone of the development is the definition of the natural logarithm in terms of an integral.<\/li>\r\n \t<li>The function [latex]{e}^{x}[\/latex] is then defined as the inverse of the natural logarithm.<\/li>\r\n \t<li>General exponential functions are defined in terms of [latex]{e}^{x},[\/latex] and the corresponding inverse functions are general logarithms.<\/li>\r\n \t<li>Familiar properties of logarithms and exponents still hold in this more rigorous context.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1167794327228\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1167793278406\">\r\n \t<li><strong>Natural logarithm function<\/strong><\/li>\r\n \t<li>[latex]\\text{ln}x={\\displaystyle\\int }_{1}^{x}\\frac{1}{t}dt[\/latex] Z<\/li>\r\n \t<li><strong>Exponential function<\/strong>[latex]y={e}^{x}[\/latex]<\/li>\r\n \t<li>[latex]\\text{ln}y=\\text{ln}({e}^{x})=x[\/latex] Z<\/li>\r\n<\/ul>\r\n<\/div>","rendered":"<div id=\"fs-id1167793423569\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1167793931233\">\n<li>The earlier treatment of logarithms and exponential functions did not define the functions precisely and formally. This section develops the concepts in a mathematically rigorous way.<\/li>\n<li>The cornerstone of the development is the definition of the natural logarithm in terms of an integral.<\/li>\n<li>The function [latex]{e}^{x}[\/latex] is then defined as the inverse of the natural logarithm.<\/li>\n<li>General exponential functions are defined in terms of [latex]{e}^{x},[\/latex] and the corresponding inverse functions are general logarithms.<\/li>\n<li>Familiar properties of logarithms and exponents still hold in this more rigorous context.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1167794327228\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1167793278406\">\n<li><strong>Natural logarithm function<\/strong><\/li>\n<li>[latex]\\text{ln}x={\\displaystyle\\int }_{1}^{x}\\frac{1}{t}dt[\/latex] Z<\/li>\n<li><strong>Exponential function<\/strong>[latex]y={e}^{x}[\/latex]<\/li>\n<li>[latex]\\text{ln}y=\\text{ln}({e}^{x})=x[\/latex] Z<\/li>\n<\/ul>\n<\/div>\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-1189\">\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 2. <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\/books\/calculus-volume-2\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/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-2\/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":29,"template":"","meta":{"_candela_citation":"{\"2\":{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/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-1189","chapter","type-chapter","status-publish","hentry"],"part":1160,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1189","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1189\/revisions"}],"predecessor-version":[{"id":2515,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1189\/revisions\/2515"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/1160"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1189\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1189"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1189"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1189"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1189"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}