Answer
Domain :$\qquad(1,+\infty)$
Range :$\qquad(-\infty,+\infty)$
Vertical asymptote: $\qquad x=1$
Work Step by Step
We start with the graph of $f_{1}(x)=\log_{3}(x)$
(dashed blue).
Since y$=\log_{3}(x-1)-2=f(x-1)-2$
its graph (red) is obtained by
1. shifting $f_{1}(x)$ right by 1 units (green dotted),
2. then shifting down by 1 units.
Domain :$\qquad(1,+\infty)$
Range :$\qquad(-\infty,+\infty)$
Vertical asymptote: $\qquad x=1$