Solve the equation: $\;$ $x^{\log_3 x} = 9$
Given equation: $\;\;$ $x^{\log_3 x} = 9$ $\;\;\; \cdots \; (1)$
Taking logarithims to the base $3$ on both sides of equation $(1)$ gives
$\log_3 \left(x^{\log_3 x}\right) = \log_3 9$
i.e. $\;$ $\log_3 x \times \log_3 x = \log_3 3^2$
i.e. $\;$ $\left(\log_3 x\right)^2 = 2 \log_3 3$
i.e. $\;$ $\left(\log_3 x\right)^2 = 2$
i.e. $\;$ $\log_3 x = \pm \sqrt{2}$
i.e. $\;$ $x = 3^{\pm \sqrt{2}}$
$\therefore \;$ The solution to the given equation is $\;\;$ $x = \left\{3^{- \sqrt{2}}, \; 3^{+ \sqrt{2}} \right\}$