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:

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

   You know that 64 = 4³, so you can say 4^{x – 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 2^{x – 5} = 8^{x – 3}, follow these steps:

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

   This step gives you 2^{x – 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 2^{x – 5} = 2^{3x – 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.