Answer
$x=\frac{1}{2}$
Work Step by Step
First, we let
$\log_{7}{\sqrt7} = x$
RECALL:
$\log_b{y} =x \longrightarrow b^x=y$
Use the rule above, where $y=\sqrt7$ and $b=7$, to obtain:
$7^x=\sqrt7$
Write $\sqrt7$ as $ 7^{\frac{1}{2}}$ to obtain:
$7^x=7^{\frac{1}{2}}$
Use the rule "If $a^x=a^y$, then $x=y$" to obtain:
$x=\frac{1}{2}$