Answer
Solution set = $\{22\}.$
Work Step by Step
For the equation to be defined, logarithms must have positive arguments
$ x -2\gt 0$
$ x \gt 2 \qquad (*)$
This is the condition eventual solutions must satisfy.
$\log(x-2)+\log 5=\log 100\qquad $... LHS: product rule
$\log(5x-10)=\log 100 \quad $ ... $\log $ is one-to-one
$5x-10=100$
$5x=110$
$ x=22\qquad $ ... satisfies (*)
Thus, the solution set = $\{22\}.$
