$$log_{216}(log_6 x) = log_6(log_{216} x)$$
$$\frac{1}{3} log_6(log_6 x) = log_6(\frac{1}{3} log_6 x)$$
$$log_6(log_6 x) = log_6(\frac{1}{3} log_6 x)^3$$
$$log_6 x = (\frac{1}{3} log_6 x)^3$$
$$log_6 x = \frac{1}{27} (log_6 x)^3$$
$$27 log_6 x = (log_6 x)^3$$
$$0 = (log_6 x)^3-27 log_6 x $$
$$0 = (log_6 x) [(log_6 x)^2-27] $$
$$\because log_6 x > 0 $$
$$\therefore (log_6 x)^2 = 27$$
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