Answer
$\approx0.3028$
Work Step by Step
Using $\log_a b=\dfrac{\log b}{\log a}$ or the Change-Of-Base Theorem, the given expression, $
\log_{\pi} \sqrt{2}
,$ evaluates to
\begin{array}{l}\require{cancel}
\dfrac{\log \sqrt{2}}{\log \pi}
\\\\
\approx0.3028
.\end{array}