{"id":2637,"date":"2025-08-13T18:19:30","date_gmt":"2025-08-13T18:19:30","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2637"},"modified":"2025-09-11T21:38:47","modified_gmt":"2025-09-11T21:38:47","slug":"exponential-and-logarithmic-equations-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-and-logarithmic-equations-background-youll-need-3\/","title":{"raw":"Exponential and Logarithmic Equations: Background You'll Need 3","rendered":"Exponential and Logarithmic Equations: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Solve linear and quadratic equations<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-start=\"267\" data-end=\"307\">Solve Linear and Quadratic Equations<\/h2>\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<div>\r\n<h3>solving equations<\/h3>\r\nLinear equations are solved with inverse operations. Quadratic equations are solved by factoring or applying the quadratic formula.\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve [latex]3x - 7 = 11[\/latex].\r\n\r\n\\begin{aligned}\r\n3x - 7 &amp;= 11 &amp;&amp; \\text{Start with the equation.} \\\\[6pt]\r\n3x &amp;= 18 &amp;&amp; \\text{Add 7 to both sides.} \\\\[6pt]\r\nx &amp;= 6 &amp;&amp; \\text{Divide both sides by 3.}\r\n\\end{aligned}\r\n\r\nThe solution is [latex]x = 6[\/latex].\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"804\" data-end=\"843\">Solve [latex]4x - 5 = 2x + 7[\/latex].<\/p>\r\n\\begin{aligned}\r\n4x - 5 &amp;= 2x + 7 &amp;&amp; \\text{Start with the equation.} \\\\[6pt]\r\n4x - 2x - 5 &amp;= 7 &amp;&amp; \\text{Subtract $2x$ from both sides.} \\\\[6pt]\r\n2x - 5 &amp;= 7 &amp;&amp; \\text{Simplify.} \\\\[6pt]\r\n2x &amp;= 12 &amp;&amp; \\text{Add 5 to both sides.} \\\\[6pt]\r\nx &amp;= 6 &amp;&amp; \\text{Divide both sides by 2.}\r\n\\end{aligned}\r\n\r\nThe solution is [latex]x = 6[\/latex].\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311421[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox recall\" aria-label=\"Recall\">A <strong data-start=\"1039\" data-end=\"1061\">quadratic equation<\/strong> has a variable raised to the second power as the highest exponent (degree of 2).\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve [latex]x^2 - 5x + 6 = 0[\/latex].\r\n\r\n\\begin{aligned}\r\nx^2 - 5x + 6 &amp;= 0 &amp;&amp; \\text{Start with the quadratic} \\\\[6pt]\r\n(x - 2)(x - 3) &amp;= 0 &amp;&amp; \\text{Factor the trinomial} \\\\[6pt]\r\nx - 2 &amp;= 0 \\quad \\text{or} \\quad x - 3 = 0 \\\\[6pt]\r\nx &amp;= 2 \\quad \\text{or} \\quad x = 3\r\n\\end{aligned}\r\n<p data-start=\"1522\" data-end=\"1588\">The solutions are [latex]x = 2[\/latex] and [latex]x = 3[\/latex].<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311429[\/ohm_question]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Solve linear and quadratic equations<\/li>\n<\/ul>\n<\/section>\n<h2 data-start=\"267\" data-end=\"307\">Solve Linear and Quadratic Equations<\/h2>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<div>\n<h3>solving equations<\/h3>\n<p>Linear equations are solved with inverse operations. Quadratic equations are solved by factoring or applying the quadratic formula.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]3x - 7 = 11[\/latex].<\/p>\n<p>\\begin{aligned}<br \/>\n3x &#8211; 7 &amp;= 11 &amp;&amp; \\text{Start with the equation.} \\\\[6pt]<br \/>\n3x &amp;= 18 &amp;&amp; \\text{Add 7 to both sides.} \\\\[6pt]<br \/>\nx &amp;= 6 &amp;&amp; \\text{Divide both sides by 3.}<br \/>\n\\end{aligned}<\/p>\n<p>The solution is [latex]x = 6[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"804\" data-end=\"843\">Solve [latex]4x - 5 = 2x + 7[\/latex].<\/p>\n<p>\\begin{aligned}<br \/>\n4x &#8211; 5 &amp;= 2x + 7 &amp;&amp; \\text{Start with the equation.} \\\\[6pt]<br \/>\n4x &#8211; 2x &#8211; 5 &amp;= 7 &amp;&amp; \\text{Subtract $2x$ from both sides.} \\\\[6pt]<br \/>\n2x &#8211; 5 &amp;= 7 &amp;&amp; \\text{Simplify.} \\\\[6pt]<br \/>\n2x &amp;= 12 &amp;&amp; \\text{Add 5 to both sides.} \\\\[6pt]<br \/>\nx &amp;= 6 &amp;&amp; \\text{Divide both sides by 2.}<br \/>\n\\end{aligned}<\/p>\n<p>The solution is [latex]x = 6[\/latex].<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311421\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311421&theme=lumen&iframe_resize_id=ohm311421&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">A <strong data-start=\"1039\" data-end=\"1061\">quadratic equation<\/strong> has a variable raised to the second power as the highest exponent (degree of 2).<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]x^2 - 5x + 6 = 0[\/latex].<\/p>\n<p>\\begin{aligned}<br \/>\nx^2 &#8211; 5x + 6 &amp;= 0 &amp;&amp; \\text{Start with the quadratic} \\\\[6pt]<br \/>\n(x &#8211; 2)(x &#8211; 3) &amp;= 0 &amp;&amp; \\text{Factor the trinomial} \\\\[6pt]<br \/>\nx &#8211; 2 &amp;= 0 \\quad \\text{or} \\quad x &#8211; 3 = 0 \\\\[6pt]<br \/>\nx &amp;= 2 \\quad \\text{or} \\quad x = 3<br \/>\n\\end{aligned}<\/p>\n<p data-start=\"1522\" data-end=\"1588\">The solutions are [latex]x = 2[\/latex] and [latex]x = 3[\/latex].<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311429\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311429&theme=lumen&iframe_resize_id=ohm311429&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":4,"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\/2637"}],"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":3,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2637\/revisions"}],"predecessor-version":[{"id":3887,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2637\/revisions\/3887"}],"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\/2637\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2637"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2637"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2637"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2637"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}