TITLE: Equation with Logarithm in Exponent
QUESTION [5 upvotes]: How to I solve the following exercise with a logarithm? I've forgotten the "trick" for doing so:
$x^{log_{10} x} =10^4$

REPLY [6 votes]: To start, take $\log_{10}$ of both sides to get:
$$\log_{10}(x)\log_{10}(x) = \log_{10}(10)^{4} = 4$$
Then just solve $$(\log_{10}(x))^{2} = 4$$
$$\implies \log_{10}x = \pm 2$$
Exponentiating both sides, we get:
$$x = 10^{2} = 100$$ OR
$$x = 10^{-2} = \frac{1}{100}$$