{"appState":{"pageLoadApiCallsStatus":true},"articleState":{"article":{"headers":{"creationTime":"2016-03-26T15:11:06+00:00","modifiedTime":"2021-07-12T15:20:09+00:00","timestamp":"2022-09-14T18:18:24+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Pre-Calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"},"slug":"pre-calculus","categoryId":33727}],"title":"How to Solve an Exponential Equation with a Variable on One or Both Sides","strippedTitle":"how to solve an exponential equation with a variable on one or both sides","slug":"how-to-solve-an-exponential-equation-with-a-variable-on-one-or-both-sides","canonicalUrl":"","seo":{"metaDescription":"Whether an exponential equation contains a variable on one or both sides, the type of equation you’re asked to solve determines the steps you take to solve it.T","noIndex":0,"noFollow":0},"content":"Whether an exponential equation contains a variable on one or both sides, the type of equation you’re asked to solve determines the steps you take to solve it.\r\n\r\nThe basic type of exponential equation has a variable on only one side and can be written with the same base for each side. For example, if you’re asked to solve 4<i><sup>x</sup></i><sup> – 2</sup> = 64, you follow these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Rewrite both sides of the equation so that the bases match.</p>\r\n<p class=\"child-para\">You know that 64 = 4<sup>3</sup>, so you can say 4<i><sup>x</sup></i><sup> – 2</sup> = 4<sup>3</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Drop the base on both sides and just look at the exponents.</p>\r\n<p class=\"child-para\">When the bases are equal, the exponents have to be equal. This step gives you the equation <i>x</i> – 2 = 3.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Solve the equation.</p>\r\n<p class=\"child-para\">This example has the solution <i>x</i> = 5.</p>\r\n</li>\r\n</ol>\r\nIf you must solve an equation with variables on both sides, you have to do a little more work (sorry!). For example, to solve 2<i><sup>x</sup></i><sup> – 5</sup> = 8<i><sup>x</sup></i><sup> – 3</sup>, follow these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Rewrite all exponential equations so that they have the same base.</p>\r\n<p class=\"child-para\">This step gives you 2<i><sup>x</sup></i><sup> – 5</sup> = (2<sup>3</sup>)<i><sup>x</sup></i><sup> – 3</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Use the properties of exponents to simplify.</p>\r\n<p class=\"child-para\">A power to a power signifies that you multiply the exponents. Distributing the exponent inside the parentheses, you get 3(<i>x</i> – 3) = 3<i>x</i> – 9, so you have 2<i><sup>x</sup></i><sup> – 5</sup> = 2<sup>3</sup><i><sup>x</sup></i><sup> – 9</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Drop the base on both sides.</p>\r\n<p class=\"child-para\">The result is <i>x</i> – 5 = 3<i>x</i> – 9.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Solve the equation.</p>\r\n<p class=\"child-para\">Subtract <i>x</i> from both sides to get –5 = 2<i>x</i> – 9. Add 9 to each side to get 4 = 2<i>x.</i> Lastly, divide both sides by 2 to get 2 = <i>x.</i></p>\r\n</li>\r\n</ol>","description":"Whether an exponential equation contains a variable on one or both sides, the type of equation you’re asked to solve determines the steps you take to solve it.\r\n\r\nThe basic type of exponential equation has a variable on only one side and can be written with the same base for each side. For example, if you’re asked to solve 4<i><sup>x</sup></i><sup> – 2</sup> = 64, you follow these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Rewrite both sides of the equation so that the bases match.</p>\r\n<p class=\"child-para\">You know that 64 = 4<sup>3</sup>, so you can say 4<i><sup>x</sup></i><sup> – 2</sup> = 4<sup>3</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Drop the base on both sides and just look at the exponents.</p>\r\n<p class=\"child-para\">When the bases are equal, the exponents have to be equal. This step gives you the equation <i>x</i> – 2 = 3.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Solve the equation.</p>\r\n<p class=\"child-para\">This example has the solution <i>x</i> = 5.</p>\r\n</li>\r\n</ol>\r\nIf you must solve an equation with variables on both sides, you have to do a little more work (sorry!). For example, to solve 2<i><sup>x</sup></i><sup> – 5</sup> = 8<i><sup>x</sup></i><sup> – 3</sup>, follow these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Rewrite all exponential equations so that they have the same base.</p>\r\n<p class=\"child-para\">This step gives you 2<i><sup>x</sup></i><sup> – 5</sup> = (2<sup>3</sup>)<i><sup>x</sup></i><sup> – 3</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Use the properties of exponents to simplify.</p>\r\n<p class=\"child-para\">A power to a power signifies that you multiply the exponents. Distributing the exponent inside the parentheses, you get 3(<i>x</i> – 3) = 3<i>x</i> – 9, so you have 2<i><sup>x</sup></i><sup> – 5</sup> = 2<sup>3</sup><i><sup>x</sup></i><sup> – 9</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Drop the base on both sides.</p>\r\n<p class=\"child-para\">The result is <i>x</i> – 5 = 3<i>x</i> – 9.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Solve the equation.</p>\r\n<p class=\"child-para\">Subtract <i>x</i> from both sides to get –5 = 2<i>x</i> – 9. Add 9 to each side to get 4 = 2<i>x.</i> Lastly, divide both sides by 2 to get 2 = <i>x.</i></p>\r\n</li>\r\n</ol>","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. 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How to Solve an Exponential Equation with a Variable on One or Both Sides

By: Mary Jane Sterling and

Updated: 07-12-2021

From The Book: Pre-Calculus Workbook For Dummies

Pre-Calculus Workbook For Dummies

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Whether an exponential equation contains a variable on one or both sides, the type of equation you’re asked to solve determines the steps you take to solve it.

The basic type of exponential equation has a variable on only one side and can be written with the same base for each side. For example, if you’re asked to solve 4^x^{– 2} = 64, you follow these steps:

Rewrite both sides of the equation so that the bases match.

You know that 64 = 4³, so you can say 4^x^{– 2} = 4³.
Drop the base on both sides and just look at the exponents.

When the bases are equal, the exponents have to be equal. This step gives you the equation x – 2 = 3.
Solve the equation.

This example has the solution x = 5.

If you must solve an equation with variables on both sides, you have to do a little more work (sorry!). For example, to solve 2^x^{– 5} = 8^x^{– 3}, follow these steps:

Rewrite all exponential equations so that they have the same base.

This step gives you 2^x^{– 5} = (2³)^x^{– 3}.
Use the properties of exponents to simplify.

A power to a power signifies that you multiply the exponents. Distributing the exponent inside the parentheses, you get 3(x – 3) = 3x – 9, so you have 2^x^{– 5} = 2³^x^{– 9}.
Drop the base on both sides.

The result is x – 5 = 3x – 9.
Solve the equation.

Subtract x from both sides to get –5 = 2x – 9. Add 9 to each side to get 4 = 2x. Lastly, divide both sides by 2 to get 2 = x.

About This Article

This article is from the book:

Pre-Calculus Workbook For Dummies ,

About the book author:

Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of Trigonometry For Dummies and Finite Math For Dummies.

This article can be found in the category:

Pre-Calculus ,