Answer
$x=4$
Work Step by Step
First, the solutions must satisfy $\left\{\begin{array}{l}
x-2\gt 0\\
x\gt 0
\end{array}\right.\quad \Rightarrow x\gt 2\qquad (*)$
in order for the equation to be defined.
LHS: Apply$ \quad\log_{a}(MN)=\log_{a}M+\log_{a}N$
RHS: Apply $\quad \log_{2}2^{3}=3$
$\log_{2}[ x(x-2)]=\log_{2}8$
... apply the principle of logarithmic equality
$x(x-2)=8$
$ x^{2}-2x-8=0\quad$to factor, find factors of $-8$ with sum $-2$
... these are $-4$ and $+2$
$(x+2)(x-4)=0$
Possible solutions:
$ x=-2\qquad$... does not satisfy (*), not a solution.
$x=4\qquad $... satisfies (*), and is a valid solution.
