Answer
$9$
Work Step by Step
Using $b^{\log_b x}=x,$ the given expression, $
7^{\log_7 9}
,$ is equal to $9$.
Alternatively, the given expression can be solved by letting $
x=7^{\log_7 9}
$.
Taking the logarithm of both sides, the equation above is equivalent to
\begin{align*}\require{cancel}
\log x&=\log7^{\log_7 9}
.\end{align*}
Using the properties of logarithms, the equation above is equivalent to
\begin{align*}\require{cancel}
\log x&=(\log_7 9)\log7^{}
&(\text{use }\log_b x^y=y\log_b x)
\\\\
\log x&=\left(\dfrac{\log9}{\log7}\right)\log7
&(\text{use Change-of-Base Formula})
\\\\
\log x&=\left(\dfrac{\log9}{\cancel{\log7}}\right)\cancel{\log7}
\\\\
\log x&=\log9
.\end{align*}
Since $\log_b x=\log_b y$ implies $x=y$, then the equation above implies
\begin{align*}\require{cancel}
x&=9
.\end{align*}
With $x=7^{\log_7 9}$ and $x=9$, then the expression $7^{\log_7 9}$ evaluates to $9$.
