Answer
The solution set is $\{3\}$.
Work Step by Step
The given equation is
$\log _2(x-1) +\log _2(x+1)=3$
Use product rule.
$\log _2 [(x-1)(x+1)]=3$
Rewrite in exponential form.
$(x-1)(x+1)= 2^3$
Use distributive property on the left.
$x^2-x+x-1=8$
Subtract $8$ from both sides.
$x^2-1-8=8-8$
Simplify.
$x^2-9=0$
Factor.
$(x-3)(x+3)=0$
Set both factors equal to $0$.
$x-3=0$ or $x+3=0$
Isolate $x$.
$x=3$ or $x=-3$
$-3$ is not in the domain of a logarithmic function.
The solution is $x=3$.