Answer
a) $12$
b) $13$
c) $4$
Work Step by Step
a) Let $x=4^{\log_4 12}.$ Taking the logarithm of both sides, then
\begin{align*}
\log x&=\log4^{\log_4 12}
.\end{align*}
Using the properties of logarithms, the equations above is equivalent to
\begin{align*}
\log x&=(\log_4 12)(\log4)
&(\text{use }\log_b x^y=y\log_b x)
.\end{align*}
Using $\log_b a=\dfrac{\log a}{\log b}$, the equation above is equivalent to
\begin{align*}\require{cancel}
\log x&=\left(\dfrac{\log12}{\cancel{\log4}}\right)(\cancel{\log4})
\\\\
\log x&=\log12
.\end{align*}
Since $\log a=\log b$ implies $a=b$, then the equation above implies
\begin{align*}\require{cancel}
x&=12
.\end{align*}
Hence, the expression $4^{\log_4 12}$ evaluates to $
12
$.
b) Using the properties of logarithms, the given expression, $
\log_9 9^{13}
$, is equivalent to
\begin{align*}
&
13\log_9 9
&(\text{use }\log_b x^y=y\log_b x)
\\&=
13(1)
&(\text{use }\log_b b=1)
\\&=
13
.\end{align*}
Hence, the expression $
\log_9 9^{13}
$ evaluates to $
13
$.
c) Using exponents, the given expression, $
\log_5 625
$, is equivalent to
\begin{align*}
&
\log_5 5^4
.\end{align*}
Using the properties of logarithms, the expression above is equivalent to
\begin{align*}
&
4\log_5 5
&(\text{use }\log_b x^y=y\log_b x)
\\&=
4(1)
&(\text{use }\log_b b=1)
\\&=
4
.\end{align*}
