Answer
The logarithmic function $f(x)=\ln{x-1}$ has the $\underline{vertical}$ asymptote $\underline{ x = 1}$.
Work Step by Step
We know that every logarithmic function has the same form (Black graph in the image below). It is like a parent function of logarithmic functions ($\ln{x}$).
Making a little transformation ($\ln{x-1}$, which means to shift the parent functions to the right by $1$ unit) we will get the Red graph on the image below. So we will have a Vertical asymptote $x=1$
