Answer
$ x=4$
Work Step by Step
$\log_{2} x+\log_{2} (x-2)=0 \implies 3=\log_{2} 2^3$
Re-arrange as: $ x(x-2)=2^3$
$\implies x^2-2x-8=0$
$\implies (x^2+2x)-(4x+8)=0$
or, $(x+2)(x-4)=0$
So, $ x=-2,4$
Neglect the negative term of $ x $, because it does not belong to the domain of the variable; so we have $ x=4$
