# How to Solve an Exponential Equation with a Variable on One or Both Sides

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 = 4

^{3}, so you can say 4^{x}^{ – 2}= 4^{3}.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}= (2^{3})^{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) = 3*x*– 9, so you have 2^{x}^{ – 5}= 2^{3}^{x}^{ – 9}.Drop the base on both sides.

The result is

*x*– 5 = 3*x*– 9.Solve the equation.

Subtract

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