Answer
$x \gt 1$ or $(1, +\infty)$
Work Step by Step
RECALL:
The logarithmic function $f(x) = \ln{x}$ is defined only when $x \gt 0$. Thus, its domain is $x \gt 0$.
This means that the function $f(x) = \ln{(x-1)}$ is defined only when $x-1 \gt 0$.
Solve the inequality to obtain:
$x-1 \gt 0
\\x-1+1 \gt 0+1
\\x \gt 1$
Thus, the domain of the given function is $x \gt 1$.
In interval notation, the domain is $(1, +\infty)$.