Answer
$x=\dfrac{8}{3}$
Work Step by Step
$125^{x-2}=25$
Rewrite $125$ as $5^{3}$ and $25$ as $5^{2}$:
$(5^{3})^{x-2}=5^{2}$
Transform the left side of the equation into a power that has the product of $3$ and $x-2$ as its exponent:
$5^{3x-6}=5^{2}$
If $5^{3x-6}=5^{2}$, then $3x-6=2$
$3x-6=2$
Solve for $x$:
$3x=2+6$
$3x=8$
$x=\dfrac{8}{3}$