Hard Logarithm Problem: $\left(\log _{10}\left(x\right)\right)^{\log _{10}\left(\log _{10}\left(x\right)\right)} = 10000$

Question

$\left(\log _{10}\left(x\right)\right)^{\log _{10}\left(\log _{10}\left(x\right)\right)} = 10000$

Solve for $x$. What I did first was to set $u = \log_{10} x$ and then try to solve for $u$. However, I got stuck a bit after will simplification and stuff like that. Any help would be appreciated.

Answer 1

Taking logs of both sides, we have

$$4 = \log_{10}(\log_{10}(x)) \cdot \log_{10}(\log_{10}(x))$$

$$ = \bigl(\log_{10}(\log_{10}(x))\bigr)^2.$$

This means

$$\log_{10}(\log_{10}(x)) = \pm2,$$

and thus,

$$x = 10^{10^{\pm2}}.$$