Answer
$-4$
Work Step by Step
Recall:
$$\log_a{b}=y \longleftrightarrow a^y=b$$
Let $y=\log_{\frac{1}{5}}{625}$
Use the definition above to obtain:
\begin{align*}
\log_{\frac{1}{5}}{625}&=y\\
\left(\frac{1}{5}\right)^y&=625\\
\left(\frac{1}{5}\right)^y&=5^4\\
\end{align*}
Use the rule $\frac{1}{a}=a^{-1}$ to obtain:
\begin{align*}
\left(5^{-1}\right)^y&=5^4\\
\end{align*}
Use the rule $\left(a^m\right)^n=a^{mn}$ to obtain:
\begin{align*}
5^{-y}&=5^4\\
\end{align*}
Use the rule $a^m=a^n\longrightarrow m=n$ to obtain:
\begin{align*}
5^{-y}&=5^4\\
-y&=4\\
y&=-4
\end{align*}
Thus, $\log_{\frac{1}{5}}{625}=-4$.
