How To Solve An Exponential Equation With A Variable On One Or ...

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HomeAcademics & The Arts ArticlesMath ArticlesPre-Calculus ArticlesHow to Solve an Exponential Equation with a Variable on One or Both SidesByMary Jane Sterling Updated2021-07-12 15:20:09From the bookPre-Calculus Workbook For DummiesShare

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Download E-BookPre-Calculus Workbook For Dummies Explore Book Pre-Calculus All-in-One For Dummies Explore BookBuy NowBuy on AmazonBuy on WileySubscribe on PerlegoDownload E-BookPre-Calculus Workbook For DummiesExplore Book Pre-Calculus All-in-One For DummiesExplore BookBuy NowBuy on AmazonBuy on WileySubscribe on PerlegoWhether 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 4x – 2 = 64, you follow these steps:

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

   You know that 64 = 43, so you can say 4x – 2 = 43.

2. 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.

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 2x – 5 = 8x – 3, follow these steps:

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

   This step gives you 2x – 5 = (23)x – 3.

2. 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 2x – 5 = 23x – 9.

3. Drop the base on both sides.

   The result is x – 5 = 3x – 9.

4. 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 (Peoria, Illinois) is the author of Algebra I For Dummies, Algebra Workbook For Dummies, Algebra II For Dummies, Algebra II Workbook For Dummies, and many other For Dummies 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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