$x^{\log x} = 10\,000$
$\log x^{\log x} = \log 10\,000$
$\log x \cdot \log x = 4$
$(\log x)^2 = 4$
$\log x = \pm 2$
$x = 10^{\pm 2}$
Check the given equation for x = 102 = 100
$x^{\log x} = 10\,000$
$100^{\log 100} = 10\,000$
$100^2 = 10\,000$ ← okay!
Check the given equation for x = 10-2 = 1/100
$x^{\log x} = 10\,000$
$\left( \frac{1}{100} \right)^{\log (\frac{1}{100})} = 10\,000$
$\left( \frac{1}{100} \right)^{-2} = 10\,000$ ← okay!
Hence, x = 100 and 1/100 answer