{"id":1467,"date":"2021-08-24T19:39:04","date_gmt":"2021-08-24T19:39:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1467"},"modified":"2022-01-31T21:58:39","modified_gmt":"2022-01-31T21:58:39","slug":"summary-review-5","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/summary-review-5\/","title":{"raw":"Summary: Review","rendered":"Summary: Review"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>Exponential functions: [latex]f(x)=a \\cdot b^{x}[\/latex] where [latex]b&gt;0, b \\neq 1[\/latex]. The value of the variable [latex]x[\/latex] can be any real number.<\/li>\r\n \t<li>Continuous growth\/decay is modeled by [latex]A(t)=a \\cdot e^{rt}[\/latex], for all real numbers [latex]r,t,[\/latex] and all positive numbers [latex]a[\/latex];\r\n<ul>\r\n \t<li>[latex]a[\/latex] is the initial value<\/li>\r\n \t<li>[latex]r[\/latex] is the continuous growth [latex](r&gt;0)[\/latex] or decay [latex](r&lt;0)[\/latex] rate per unit time<\/li>\r\n \t<li>and [latex]t[\/latex] is the elapsed time.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>One-to-one property of exponential functions: [latex]b^{x}=b^{y}[\/latex] if and only if [latex]x=y[\/latex], where [latex]b&gt;0, b \\neq 1[\/latex].<\/li>\r\n \t<li>Logarithmic function with base [latex]b[\/latex]: For [latex]b&gt;0, b \\neq 1, y = \\mathrm{log}_{b} \\ x[\/latex] if and only if [latex]b^{y}=x[\/latex], where [latex]x&gt;0[\/latex].\r\n<ul>\r\n \t<li>[latex]y=\\mathrm{log}_b \\ x[\/latex] os the\u00a0logarithmic form<\/li>\r\n \t<li>[latex]b^{y} = x[\/latex] is the\u00a0exponential form<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Exponential\u00a0equations may be solve by\r\n<ul>\r\n \t<li>The one-to-one property: [latex]b^{S}=b^{T}[\/latex] if and only if [latex]S=T[\/latex], or<\/li>\r\n \t<li>By isolating the exponential expression and writing in logarithmic form. Then solve for the variable.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<ul>\r\n \t<li aria-level=\"1\"><strong>common logarithms:\u00a0<\/strong>have an implied base [latex]b=10[\/latex]: [latex]\\mathrm{log}(x) = \\mathrm{log}_{10} (x)[\/latex<\/li>\r\n \t<li aria-level=\"1\"><strong>continuous random variables: <\/strong>variables that can take on any value within a range of values<\/li>\r\n \t<li aria-level=\"1\">[latex]\\boldsymbol{e}[\/latex]: the irrational number which is the limiting value of [latex](1+ \\frac{1}{n})^{n}[\/latex] as [latex]n[\/latex] increases without bound, [latex]e \\approx 2.718282[\/latex]<\/li>\r\n \t<li aria-level=\"1\"><strong>exponential growth<\/strong>: quantity grows by a rate proportional to the current amount<\/li>\r\n \t<li aria-level=\"1\"><strong>natural\u00a0<\/strong><b>logarithms:\u00a0<\/b>have base [latex]e[\/latex]: [latex]\\mathrm{ln} (x) = \\mathrm{log}_{e} (x)[\/latex]<\/li>\r\n<\/ul>","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li>Exponential functions: [latex]f(x)=a \\cdot b^{x}[\/latex] where [latex]b>0, b \\neq 1[\/latex]. The value of the variable [latex]x[\/latex] can be any real number.<\/li>\n<li>Continuous growth\/decay is modeled by [latex]A(t)=a \\cdot e^{rt}[\/latex], for all real numbers [latex]r,t,[\/latex] and all positive numbers [latex]a[\/latex];\n<ul>\n<li>[latex]a[\/latex] is the initial value<\/li>\n<li>[latex]r[\/latex] is the continuous growth [latex](r>0)[\/latex] or decay [latex](r<0)[\/latex] rate per unit time<\/li>\n<li>and [latex]t[\/latex] is the elapsed time.<\/li>\n<\/ul>\n<\/li>\n<li>One-to-one property of exponential functions: [latex]b^{x}=b^{y}[\/latex] if and only if [latex]x=y[\/latex], where [latex]b>0, b \\neq 1[\/latex].<\/li>\n<li>Logarithmic function with base [latex]b[\/latex]: For [latex]b>0, b \\neq 1, y = \\mathrm{log}_{b} \\ x[\/latex] if and only if [latex]b^{y}=x[\/latex], where [latex]x>0[\/latex].\n<ul>\n<li>[latex]y=\\mathrm{log}_b \\ x[\/latex] os the\u00a0logarithmic form<\/li>\n<li>[latex]b^{y} = x[\/latex] is the\u00a0exponential form<\/li>\n<\/ul>\n<\/li>\n<li>Exponential\u00a0equations may be solve by\n<ul>\n<li>The one-to-one property: [latex]b^{S}=b^{T}[\/latex] if and only if [latex]S=T[\/latex], or<\/li>\n<li>By isolating the exponential expression and writing in logarithmic form. Then solve for the variable.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<ul>\n<li aria-level=\"1\"><strong>common logarithms:\u00a0<\/strong>have an implied base [latex]b=10[\/latex]: [latex]\\mathrm{log}(x) = \\mathrm{log}_{10} (x)[\/latex<\/li>\n<li aria-level=\"1\"><strong>continuous random variables: <\/strong>variables that can take on any value within a range of values<\/li>\n<li aria-level=\"1\">[latex]\\boldsymbol{e}[\/latex]: the irrational number which is the limiting value of [latex](1+ \\frac{1}{n})^{n}[\/latex] as [latex]n[\/latex] increases without bound, [latex]e \\approx 2.718282[\/latex]<\/li>\n<li aria-level=\"1\"><strong>exponential growth<\/strong>: quantity grows by a rate proportional to the current amount<\/li>\n<li aria-level=\"1\"><strong>natural\u00a0<\/strong><b>logarithms:\u00a0<\/b>have base [latex]e[\/latex]: [latex]\\mathrm{ln} (x) = \\mathrm{log}_{e} (x)[\/latex]<\/li>\n<\/ul>\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-1467\">\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>College Algebra. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites\">https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites<\/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>: Access for free at https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites<\/li><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/precalculus\/pages\/1-introduction-to-functions\">https:\/\/openstax.org\/books\/precalculus\/pages\/1-introduction-to-functions<\/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>: Access for free at https:\/\/openstax.org\/books\/precalculus\/pages\/1-introduction-to-functions<\/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":169134,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/precalculus\/pages\/1-introduction-to-functions\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/precalculus\/pages\/1-introduction-to-functions\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1467","chapter","type-chapter","status-publish","hentry"],"part":249,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1467","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":8,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1467\/revisions"}],"predecessor-version":[{"id":3606,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1467\/revisions\/3606"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/249"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1467\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1467"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1467"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1467"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1467"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}