Answer
$4$
Work Step by Step
Apply the logarithmic property:
$\log_a M +\log_a N = \log_a (M \ N )$
and rearrange the given expression to obtain:
$\log_a [x(x-2)]= \log_a (x-4).(1)$
When $\log_a M=\log_a N$, then we have: $M=N$
Now, equation (1) becomes:
$x(x-2)=x-4 \implies x^2-2x=x-4 \implies x^2 -3x+4=0 $
This gives a quadratic equation, whose factors are:
$(x-4)(x+1)=0$
Use the zero factor property to obtain:
$ x-4 =0 \implies x=4$ and $x+1=0 \implies x=-1$
Since the domain of the variable is $x \gt 0$, we will have to discard $x=-1$. Thus, our answer is: $x=4$
