How do i get from $x^{(\log(x))}=10000 $ to $\log(x)^2=\log(10000)$

Question

I'm looking at the solution for a math problem I'm trying to solve and can't comprehend the following step:

From: $$ x^{\log_{10}(x)}=10000 $$ To: $$ {\log_{10}(x)}^2=\log_{10}(10000) $$ Is there a specific rule how this step works or is it just some kind of logarithmic logic?

Answer 1

Hint: $$x^{\log_{10}(x)}= (10^{\log_{10}(x)})^{\log_{10}(x)} = 10^{\log_{10}(x)\cdot\log_{10}(x)}$$

Can you take it form here?

Answer 2

Specfic rule is the power law $$ a^b = c\,\,\text{taking logs}\\ \log \left(a^b\right) = b\log (a) = \log c $$ now if you had $b = \log_{10} (a)$ what do you get?