Solve the equation: $\;$ $x^{\log_{\sqrt{x}} \left(x - 2\right)} = 9$
Given equation: $\;\;$ $x^{\log_{\sqrt{x}} \left(x - 2\right)} = 9$ $\;\;\;\; \cdots \; (1)$
Taking log to base $\sqrt{x}$ on either side of equation $(1)$ gives
$\log_{\sqrt{x}} \left[x^{\log_{\sqrt{x}}\left(x - 2\right)}\right] = \log_{\sqrt{x}} 9$
i.e. $\;$ $\log_{\sqrt{x}} \left(x - 2\right) \log_{\sqrt{x}} x = \log_{\sqrt{x}} 9$
i.e. $\;$ $\log_{\sqrt{x}} \left(x-2\right) \log_{\sqrt{x}} \left(\sqrt{x}\right)^2 = \log_{\sqrt{x}} 3^2$
i.e. $\;$ $2 \log_{\sqrt{x}} \left(x-2\right) \log_{\sqrt{x}} \sqrt{x} = 2\log_{\sqrt{x}} 3$
i.e. $\;$ $\log_{\sqrt{x}} \left(x-2\right) = \log_{\sqrt{x}} 3$ $\;\;\; \cdots \; (2)$
Taking anti-log to base $\sqrt{x}$ on either side of equation $(2)$ gives
$x - 2 = 3$ $\implies$ $x = 5$
$\therefore \;$ The solution to the given equation is $\;\;$ $x = \left\{5 \right\}$