Answer: 27

Small Hint:

Set $y=\log_2x$ and rewrite every base-$8$ logarithm in base $2$

Big Hint:

Solve the resulting linear equation for $\log_2 y$

Solution:

Put $y=\log_2x\gt0.$ Since $\log_8x=\frac{y}{3},$ the equation becomes $$\log_2(\frac{y}{3})=\frac13\log_2y.$$ Hence $\frac23\log_2y=\log_23,$ so $y=3^{\frac{3}{2}}=3\sqrt3.$ Therefore $(\log_2x)^2=y^2=27.$