Answer
$x\displaystyle \in\{\frac{1}{100,000},\quad 100,000\}$
Work Step by Step
In order for the equation to be defined,
$\left\{\begin{array}{l}
x\gt 0\\
x^{\log x}\gt 0
\end{array}\right.\qquad(*)$
apply $ \quad\log_{a}M^{p}=p\cdot\log_{a}M$
$\log x\cdot\log x=25\qquad$ ... let $t=\log x$
$t^{2}=25$
$t=\pm 5$
$\left[\begin{array}{lll}
\log x=-5 & ...or... & \log x=5\\
x=10^{-5} & & x=10^{5}
\end{array}\right]$
Both satisfy (*), so they are both valid solutions
$(10^{-5})^{\log 10^{-5}}=(10^{-5})^{-5}=10^{25}\gt 0$
$(10^{5})^{\log 10^{5}}=(10^{5})^{5}=10^{25}\gt 0$
