An exponential equation is one with exponents in which the exponent (or a part of the exponent) is a variable. Exponential equations are classified into three categories:

- with the same bases on both sides: 4^x = 4^2
- with different bases can be made the same: 4^x= 16 which can be written as 4^x = 42
- with different bases that cannot be made the same: 4^x = 15

When the bases on both sides of an exponential equation are equal, then the exponents must also be equal.

Sometimes, though the exponents on both sides are different, they can be made the same. For instance, 5^x = 125 (as 125 = 53).

In each of these circumstances, we have to apply the property of equality of exponential equations, which allows us to set the exponents to the same value and solve for the variable.

## Example:

Solve $7^{y+1} = 343^y$

### Solution:

We know that 343 = 7^3, so we can write this equation this way:

$7^{y+1} = 7^3y$

$7^{y+1} = 7^{3y}$

Given that both sides are the same, we can say that their exponents are the same as well. So, y + 1 = 3y.

Subtracting y from both sides, 2y = 1. Dividing both sides by 2, y = ½.

Answer: y = ½