Answer
$x=\frac{8}{3}$
Work Step by Step
We are given that $125^{x-2}=25$. Both of these numbers are powers of 5, so we can rewrite the equation as $(5^{3})^{x-2}=5^{3x-6}=5^{2}$.
From the uniqueness of $b^{x}$, we know that $b^{x}=b^{y}$ is equivalent to $x=y$ (when $b\gt0$ and $b\ne1$).
Therefore, $3x-6=2$.
Add 6 to both sides.
$3x=8$
Divide both sides by 3.
$x=\frac{8}{3}$
