{"id":2634,"date":"2025-08-13T18:19:15","date_gmt":"2025-08-13T18:19:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2634"},"modified":"2025-09-11T20:24:57","modified_gmt":"2025-09-11T20:24:57","slug":"exponential-and-logarithmic-equations-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-and-logarithmic-equations-background-youll-need-2\/","title":{"raw":"Exponential and Logarithmic Equations: Background You'll Need 2","rendered":"Exponential and Logarithmic Equations: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Write numbers in exponential form<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-start=\"401\" data-end=\"440\">Writing Numbers in Exponential Form<\/h2>\r\n<p data-start=\"442\" data-end=\"590\">When solving exponential equations, it\u2019s often helpful to <strong data-start=\"500\" data-end=\"546\">rewrite numbers as powers of the same base<\/strong>. This lets us compare exponents directly.<\/p>\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">\r\n<p data-start=\"618\" data-end=\"669\">Some numbers are perfect powers of smaller bases.<\/p>\r\n\r\n<ul data-start=\"671\" data-end=\"738\">\r\n \t<li data-start=\"671\" data-end=\"686\">\r\n<p data-start=\"673\" data-end=\"686\">[latex]8=2^3[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"687\" data-end=\"703\">\r\n<p data-start=\"689\" data-end=\"703\">[latex]81=9^2=3^4[\/latex]<\/p>\r\n\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn><\/mn><\/mrow><\/semantics><\/math><\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311416[\/ohm_question]<\/section><section class=\"textbox recall\" aria-label=\"Recall\">\r\n<p data-start=\"947\" data-end=\"1026\">Fractions are often powers of whole numbers, but with <strong data-start=\"1001\" data-end=\"1023\">negative exponents<\/strong>.<\/p>\r\n<p style=\"text-align: center;\" data-start=\"947\" data-end=\"1026\">[latex]\\frac{1}{a^n}=a^{-n}[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Rewrite [latex]\\frac{1}{25}[\/latex] in exponential form.[reveal-answer q=\"774185\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"774185\"]Start by identifying the exponential form of [latex]25[\/latex].Since [latex]25=5^2[\/latex] we know [latex]\\frac{1}{25}=\\frac{1}{5^2}[\/latex].\r\n\r\nNow, rewrite the fraction with a negative exponent.\r\n\r\n[latex]\\frac{1}{5^2}=5^{-2}[\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311417[\/ohm_question]<\/section><section class=\"textbox recall\" aria-label=\"Recall\"><strong>Power Property of Exponents<\/strong>When raising a power to another power, multiply the exponents:\r\n\r\n[latex](a^m)^n=a^{m*n}[\/latex]<\/section><section class=\"textbox example\" aria-label=\"Example\">Rewrite [latex]25^x[\/latex] using base [latex]5[\/latex].[reveal-answer q=\"314075\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"314075\"]\\begin{aligned}\r\n25^x &amp;= (5^2)^x &amp;&amp; \\text{Rewrite 25 as } 5^2 \\\\[6pt]\r\n&amp;= 5^{2x} &amp;&amp; \\text{Apply the power property}\r\n\\end{aligned}[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Rewrite [latex]9^{x+1}[\/latex] using base [latex]3[\/latex][reveal-answer q=\"641212\"]Show Solutions[\/reveal-answer]\r\n[hidden-answer a=\"641212\"]\\begin{aligned} 9^{x+1} &amp;= (3^2)^{x+1} &amp;&amp; \\text{Rewrite 9 as } 3^2 \\\\[6pt] &amp;= 3^{2(x+1)} &amp;&amp; \\text{Apply the power property} \\\\[6pt] &amp;= 3^{2x+2} &amp;&amp; \\text{Distribute the multiplication} \\end{aligned}[\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311418[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Write numbers in exponential form<\/span><\/li>\n<\/ul>\n<\/section>\n<h2 data-start=\"401\" data-end=\"440\">Writing Numbers in Exponential Form<\/h2>\n<p data-start=\"442\" data-end=\"590\">When solving exponential equations, it\u2019s often helpful to <strong data-start=\"500\" data-end=\"546\">rewrite numbers as powers of the same base<\/strong>. This lets us compare exponents directly.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<p data-start=\"618\" data-end=\"669\">Some numbers are perfect powers of smaller bases.<\/p>\n<ul data-start=\"671\" data-end=\"738\">\n<li data-start=\"671\" data-end=\"686\">\n<p data-start=\"673\" data-end=\"686\">[latex]8=2^3[\/latex]<\/p>\n<\/li>\n<li data-start=\"687\" data-end=\"703\">\n<p data-start=\"689\" data-end=\"703\">[latex]81=9^2=3^4[\/latex]<\/p>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn><\/mn><\/mrow><\/semantics><\/math><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311416\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311416&theme=lumen&iframe_resize_id=ohm311416&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<p data-start=\"947\" data-end=\"1026\">Fractions are often powers of whole numbers, but with <strong data-start=\"1001\" data-end=\"1023\">negative exponents<\/strong>.<\/p>\n<p style=\"text-align: center;\" data-start=\"947\" data-end=\"1026\">[latex]\\frac{1}{a^n}=a^{-n}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Rewrite [latex]\\frac{1}{25}[\/latex] in exponential form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q774185\">Show Solution<\/button><\/p>\n<div id=\"q774185\" class=\"hidden-answer\" style=\"display: none\">Start by identifying the exponential form of [latex]25[\/latex].Since [latex]25=5^2[\/latex] we know [latex]\\frac{1}{25}=\\frac{1}{5^2}[\/latex].<\/p>\n<p>Now, rewrite the fraction with a negative exponent.<\/p>\n<p>[latex]\\frac{1}{5^2}=5^{-2}[\/latex]<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311417\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311417&theme=lumen&iframe_resize_id=ohm311417&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\"><strong>Power Property of Exponents<\/strong>When raising a power to another power, multiply the exponents:<\/p>\n<p>[latex](a^m)^n=a^{m*n}[\/latex]<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Rewrite [latex]25^x[\/latex] using base [latex]5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q314075\">Show Solution<\/button><\/p>\n<div id=\"q314075\" class=\"hidden-answer\" style=\"display: none\">\\begin{aligned}<br \/>\n25^x &amp;= (5^2)^x &amp;&amp; \\text{Rewrite 25 as } 5^2 \\\\[6pt]<br \/>\n&amp;= 5^{2x} &amp;&amp; \\text{Apply the power property}<br \/>\n\\end{aligned}<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Rewrite [latex]9^{x+1}[\/latex] using base [latex]3[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q641212\">Show Solutions<\/button><\/p>\n<div id=\"q641212\" class=\"hidden-answer\" style=\"display: none\">\\begin{aligned} 9^{x+1} &amp;= (3^2)^{x+1} &amp;&amp; \\text{Rewrite 9 as } 3^2 \\\\[6pt] &amp;= 3^{2(x+1)} &amp;&amp; \\text{Apply the power property} \\\\[6pt] &amp;= 3^{2x+2} &amp;&amp; \\text{Distribute the multiplication} \\end{aligned}<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311418\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311418&theme=lumen&iframe_resize_id=ohm311418&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":510,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2634"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2634\/revisions"}],"predecessor-version":[{"id":3885,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2634\/revisions\/3885"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/510"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2634\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2634"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2634"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2634"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2634"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}