{"id":94,"date":"2022-04-12T12:05:52","date_gmt":"2022-04-12T12:05:52","guid":{"rendered":"http:\/\/8thclass.deltapublications.in\/?page_id=94"},"modified":"2025-10-07T06:36:44","modified_gmt":"2025-10-07T06:36:44","slug":"f-3-solve-equations-with-variable-exponents","status":"publish","type":"page","link":"https:\/\/8thclass.deltapublications.in\/index.php\/f-3-solve-equations-with-variable-exponents\/","title":{"rendered":"F.3 Solve equations with variable exponents"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong>Solve equations with variable exponents<\/strong><\/h2>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-09eb8d27d0d70f35bd20b1cf47293f11\" style=\"color:#74008b\"><strong>Key Notes :<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table has-normal-font-size\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td><strong>\u2728 Key Idea:<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>An <strong>exponential equation<\/strong> is an equation where the <strong>unknown variable is in the exponent<\/strong>.<br>Example: 2<sup>x<\/sup>=8<\/p>\n\n\n\n<p>We solve such equations by making <strong>bases the same<\/strong> or by using <strong>logarithms<\/strong> (at higher levels).<\/p>\n\n\n\n<figure class=\"wp-block-table has-normal-font-size\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td><strong><strong>\ud83d\udd39 Step 1: Rewrite with the Same Base<\/strong><\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If both sides can be written with the same base, equate the exponents.<\/li>\n<\/ul>\n\n\n\n<p><strong>Example 1:<\/strong>  <\/p>\n\n\n\n<p class=\"has-text-align-center\">2<sup>x<\/sup>=8<\/p>\n\n\n\n<p>Write 8 as 2<sup>3<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center\">2<sup>x<\/sup>=2<sup>3<\/sup><\/p>\n\n\n\n<p>Since the bases are the same: <\/p>\n\n\n\n<p class=\"has-text-align-center\">x=3<\/p>\n\n\n\n<figure class=\"wp-block-table has-normal-font-size\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td><strong><strong>\ud83d\udd39 Step 2: Use Properties of Exponents<\/strong><\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>Remember:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>a<sup>m<\/sup> = a<sup>n<\/sup>\u2005\u200a\u27f9\u2005\u200am = n (if a&gt;0and a\u22601)<\/li>\n\n\n\n<li>(a<sup>m<\/sup>)<sup>n<\/sup> = a<sup>m\u22c5n<\/sup><\/li>\n\n\n\n<li>a<sup>m<\/sup>\u22c5a<sup>n<\/sup> = a<sup>m+n<\/sup><\/li>\n<\/ul>\n\n\n\n<p><strong>Example 2:<\/strong> <\/p>\n\n\n\n<p class=\"has-text-align-center\">3<sup>2x<\/sup> = 27<\/p>\n\n\n\n<p>Write 27 as 3<sup>3<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center\">3<sup>2x<\/sup> = 3<sup>3<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center\">2x = 3<\/p>\n\n\n\n<p class=\"has-text-align-center\"> x = 3\/2<\/p>\n\n\n\n<figure class=\"wp-block-table has-normal-font-size\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td><strong><strong>\ud83d\udd39 Step 3: When Bases Are Different<\/strong><\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If bases cannot be rewritten to be the same, we use <strong>logarithms<\/strong> (optional for grade 8, but good to introduce).<\/li>\n<\/ul>\n\n\n\n<p><strong>Example 3:<\/strong> <\/p>\n\n\n\n<p class=\"has-text-align-center\">5x = 12<\/p>\n\n\n\n<p>Take log on both sides: <\/p>\n\n\n\n<p class=\"has-text-align-center\">x\u22c5log\u20615 = log\u206112<\/p>\n\n\n\n<p class=\"has-text-align-center\">   x = log\u206112\/log\u20615x <\/p>\n\n\n\n<figure class=\"wp-block-table has-normal-font-size\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td><strong><strong>\ud83d\udd39 Step 4: Equations with Exponential Expressions on Both Sides<\/strong><\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><strong>Example 4:<\/strong> <\/p>\n\n\n\n<p class=\"has-text-align-center\">4<sup>x+1<\/sup> = 16<sup>x\u22122<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center\">4<sup>x+1<\/sup> = (4<sup>2<\/sup>)<sup>x-2<\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center\">4<sup>x+1<\/sup> =  4<sup>2x-4 <\/sup><\/p>\n\n\n\n<p class=\"has-text-align-center\">x+1 = 2x \u2212 4<\/p>\n\n\n\n<p class=\"has-text-align-center\">x = 5<\/p>\n\n\n\n<figure class=\"wp-block-table has-normal-font-size\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td><strong><strong>\u2705 Summary \/ Key Points<\/strong><\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Rewrite numbers with the <strong>same base<\/strong> whenever possible.<\/li>\n\n\n\n<li>If bases are the same \u2192 equate exponents.<\/li>\n\n\n\n<li>Use laws of exponents to simplify.<\/li>\n\n\n\n<li>If bases are different and cannot be matched \u2192 use <strong>logarithms<\/strong>.<\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-table has-normal-font-size\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td><strong><strong>\ud83d\udcdd Practice Problems<\/strong><\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<ol class=\"wp-block-list\">\n<li>2<sup>x<\/sup>=32<\/li>\n\n\n\n<li>7<sup>2x<\/sup>=343<\/li>\n\n\n\n<li>9<sup>x+1<\/sup>=81<sup>x\u22122<\/sup><\/li>\n\n\n\n<li>3<sup>x=<\/sup>20<\/li>\n\n\n\n<li>5<sup>2x\u22121<\/sup>=25<sup>x+2<\/sup><\/li>\n<\/ol>\n\n\n\n<p class=\"has-text-align-center has-text-color has-large-font-size\" style=\"color:#105000\"><strong>Learn with an example<\/strong><\/p>\n\n\n\n<div class=\"wp-block-group has-primary-color has-text-color has-background has-link-color has-normal-font-size wp-elements-8a24ad009a7be9e5ce0b49ac1eaa3eb8\" style=\"background-color:#c9ffd6\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-text-color has-link-color wp-elements-cc24e1b8fbe3344fe99ffcf513870a71\" style=\"color:#b00012\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-text-color has-link-color has-normal-font-size wp-elements-ebdf52fff82accf5e1a2926fbbf7a2b1\" style=\"color:#d90000\"><strong>Solve for u.<\/strong><\/p>\n\n\n\n<p class=\"has-primary-color has-text-color has-link-color has-normal-font-size wp-elements-1f07d35aaf6d55b7d4deefa23dcb4c77\"> 2<sup>c<\/sup> = 4<br> u =<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-primary-color has-background-background-color has-text-color has-background has-link-color wp-elements-78e5545d099645b32283dbb676c633c0\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p>Look at the equation.<br>2<sup>c<\/sup> = 4<\/p>\n\n\n\n<p>The base is 2. The exponent, u, is the number of times 2 is used in the repeated multiplication.<\/p>\n\n\n\n<p>Compute powers of 2 until you get 4.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"643\" height=\"227\" src=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2024\/08\/image-13.png\" alt=\"\" class=\"wp-image-17301\" srcset=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2024\/08\/image-13.png 643w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2024\/08\/image-13-300x106.png 300w\" sizes=\"auto, (max-width: 643px) 100vw, 643px\" \/><\/figure>\n\n\n\n<p>So, u = 2 .<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-primary-color has-text-color has-background has-link-color has-large-font-size wp-elements-568c709ae0952f80b2b0c10b091a54f4\" style=\"background-color:#ffc9fa\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-text-color has-link-color has-large-font-size wp-elements-9e6c382b9b7c80d618b1c4dd4d118356\" style=\"color:#b00012\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-normal-font-size\"><strong>Solve&nbsp;for&nbsp;u<\/strong><\/p>\n\n\n\n<p class=\"has-primary-color has-text-color has-link-color has-normal-font-size wp-elements-cb13b1e8f3e0f289a8dba20c27a2300a\">2<sup>u<\/sup> = 8<br>u =<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-primary-color has-background-background-color has-text-color has-background has-link-color has-normal-font-size wp-elements-552cfdcb2ecf85be3bbae6886db67ecf\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p class=\"has-normal-font-size\">Look&nbsp;at the&nbsp;equation.<\/p>\n\n\n\n<p>2<sup>u<\/sup> = 8<\/p>\n\n\n\n<p class=\"has-normal-font-size\">The&nbsp;base is&nbsp;2.&nbsp;The exponent,&nbsp;u,&nbsp;is the number of times&nbsp;2&nbsp;is used in the repeated&nbsp;multiplication.<\/p>\n\n\n\n<p class=\"has-normal-font-size\">Compute&nbsp;powers of&nbsp;2&nbsp;until you get&nbsp;8.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"600\" height=\"300\" src=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2022\/10\/Untitled-design-8.png\" alt=\"\" class=\"wp-image-2878\" srcset=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2022\/10\/Untitled-design-8.png 600w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2022\/10\/Untitled-design-8-300x150.png 300w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/figure>\n\n\n\n<p class=\"has-normal-font-size\">So,&nbsp;u = 3.<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<p class=\"has-text-color has-large-font-size\" style=\"color:#d90000\"><strong>Let&#8217;s practice!\ud83d\udd8a\ufe0f<\/strong><\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/77129\/164\/991\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-295.png\" alt=\"\" class=\"wp-image-9577\" srcset=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-295.png 500w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-295-300x300.png 300w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-295-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/77038\/555\/380\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-302.png\" alt=\"\" class=\"wp-image-9578\" srcset=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-302.png 500w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-302-300x300.png 300w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-302-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Solve equations with variable exponents Key Notes : \u2728 Key Idea: An exponential equation is an equation where the unknown variable is in the exponent.Example: 2x=8 We solve such equations by making bases the same or by using logarithms (at higher levels). \ud83d\udd39 Step 1: Rewrite with the Same Base Example 1: 2x=8 Write 8<a class=\"more-link\" href=\"https:\/\/8thclass.deltapublications.in\/index.php\/f-3-solve-equations-with-variable-exponents\/\">Continue reading <span class=\"screen-reader-text\">&#8220;F.3 Solve equations with variable exponents&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-94","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_hostinger_reach_plugin_has_subscription_block":false,"_hostinger_reach_plugin_is_elementor":false,"_links":{"self":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/94","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=94"}],"version-history":[{"count":33,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/94\/revisions"}],"predecessor-version":[{"id":21748,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/94\/revisions\/21748"}],"wp:attachment":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=94"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}