Answer
$4$
Work Step by Step
Recall:
$$\log_a{b}=y \longleftrightarrow a^y=b$$
Let $y=\log_4{256}$
Use the definition above to obtain:
\begin{align*}
\log_4{256}&=y\\
4^y&=256\\
4^y&=4^4
\end{align*}
Use the rule $a^m=a^n\longrightarrow m=n$ to obtain:
\begin{align*}
4^y&=4^4\\
y&=4
\end{align*}
Thus, $\log_4{256}=4$.