Answer
Please see below.
Work Step by Step
The graph of $g(x)=\log (x-1)$ (the red graph) is the graph of $f(x)=\log x$ (the blue graph) shifted one unit to the right.
Looking at the graph, we observe that the vertical asymptote is $x=1$.
We also find that the domain and the range of the function $g(x)$ are$$D_g=(1, \infty ) \quad \text{ and } \quad R_g= (- \infty, \infty ),$$respectively.
