Logarithmic Equation

Using the Properties of Logarithms , this equation can be written as

$\log_{2}((\frac{1}{3})\log_{2} x) = (\frac{1}{3})\log_{2}(\log_{2} x)$

$\Longrightarrow (\frac{1}{3})\log_{2} x = (\log_{2} x)^{\frac{1}{3}}$

$\Longrightarrow (\log_{2} x)^{\frac{2}{3}} = 3 \Longrightarrow \log_{2} x = 3\sqrt{3}$ ,

the square of which is $9 \cdot 3 = \boxed{27}$ .

$\log_2(\log_8x)=\log_8(\log_2x)$

$\implies\log_2\left(\frac{\log_2x}{3}\right)=\frac{\log_2(\log_2x)}{3}$

$\implies\log_2\left(\frac{\log_2x}{3}\right)=\log_2\left((\log_2x)^{1/3}\right)$

$\implies\frac{\log_2x}{3}=(\log_2x)^{1/3}$

$\implies\frac{(\log_2x)^3}{27}=\log_2x$

$\implies(\log_2x)^2=\boxed{27}.$

${log}_{2}(\frac{1}{3} {log}_{2}x) = \frac{1}{3}{log}{2}({log}_{2}x)$

$3{log}_{2}\frac{1}{3} + 3{log}_{2}({log}_{2}x) = {log}_{2}({log}_{2}x)$

$2{log}_{2}({log}_{2}x) = {log}_{2} 27$

${{log}_{2}x}^{2} = {2}^{{log}_{2} 27}$

${{log}_{2}x}^{2} = 27$

I can see why some find this somewhat tricky at first.

log((1/3) log(x)) = 1/3 log(log(x))

(1/3)*log(x) = (log(x))^(1/3)

(log(x))^3 = 27 (log(x))

(log(x))^2 = 27