The Rule:

If two powers are equal and their bases are the same, then their exponents must be equal.

(As long as $a$ is not -1, 0, or 1)

Real-World Example:

Let’s solve for $x$:

Step 1: Make the bases match.

We have a base of $2$ on the left. We need to turn $128$ into a power of $2$.

If we test our powers of 2, we find that $2^7 = 128$.

So, we can rewrite the equation as:

Step 2: Drop the bases.

Since the bases (2) are identical, we can ignore them and just set the exponents equal to each other.

Step 3: Solve for x.

Subtract 7 from both sides.