Answer
$x=\frac{4}{3}$
Work Step by Step
Note that:
$125=5^3$
AND
$625=5^4$
Write each side with a base of $5$ to obtain:
$(5^3)^{x}=5^4
\\5^{3x}=5^4$
RECALL:
If $a^x = a^y$, then $x=y$.
Use the rule above to obtain:
$5^{3x}=5^4
\\3x=4
\\x=\frac{4}{3}$