Log and Log

$\begin{aligned} \log_{216}\left(\log_6 x\right) & = \log_6 \left( \log_{216} x \right) \\ \frac{\log_{6}\left(\log_6 x\right)}{\log_6 216} & = \log_6 \left( \frac{\log_{6} x }{\log_6 216} \right) \\ \frac{\log_{6}\left(\log_6 x\right)}{3} & = \log_6 \left( \frac{\log_{6} x }{3} \right) \\ \log_{6}\left(\log_6 x\right) & = 3\log_6 \left( \frac{\log_{6} x }{3} \right) \\ \log_{6}\left(\log_6 x\right) & = \log_6 \left( \frac{\log_{6} x }{3} \right)^3 \\ \log_6 x & = \frac{\left(\log_{6} x \right)^3}{27} \\ \Rightarrow \left(\log_{6} x \right)^2 & = \boxed{27} \end{aligned}$

$\log_{216}(\log_{6}x) = \log_{6}(\log_{216}x) \Rightarrow \log_{6^3}(\log_{6}x) = \log_{6}(\log_{6^3}x)$

Using property of logarithms: $\log_{c^d}a = \dfrac{1}{d}\log_{c}a$

$\Rightarrow \dfrac{1}{3}\log_{6}(\log_{6}x) = \log_{6}(\dfrac{1}{3}\log_{6}x) = \log_{6}(\log_{6}x) - \log_{6}3$

$\Rightarrow \log_{6}(3\sqrt{3}) = \log_{6}(\log_{6}x) \Rightarrow \log_{6}x = 3\sqrt{3}$

$\Rightarrow (\log_{6}x)^2 = \boxed{27}$