# Solve for x, given x^((log_5 x)-2) = (x^2/(125)) ?

Oct 23, 2017

$x = \left\{5 , 125\right\}$

#### Explanation:

Applying ${\log}_{5}$ to both sides

$\left({\log}_{5} x - 2\right) {\log}_{5} x = 2 {\log}_{5} x - 3$ and now calling $y = {\log}_{5} x$

$\left(y - 2\right) y = 2 y - 3$ or ${y}^{2} - 4 y + 3 = 0$ now solving for $y$

$y = \left\{1 , 3\right\}$ or

${\log}_{5} x = 1 \Rightarrow x = 5$ and ${\log}_{5} x = 3 \Rightarrow x = 125$