Answer
The domain: $x\gt1$.
The x-intercept: $(1.002,1)$.
The vertical asymptote: $x=1$.
See graph
Work Step by Step
$$f(x)=\log_5(x-1)+4$$ The function is defined when:
$$x-1\gt0$$ $$x\gt1$$ Thus, the domain is $x\gt1$.
Finding the $x$-intercept, set $f(x)=0$:
$$0=\log_5(x-1)+4$$ $$\log_5(x-1)=-4$$ Rewriting in exponential form:
$$x-1=5^{-4}$$ $$x=\frac{1}{5^4}+1$$ $$x=1.002$$
Thus, the x-intercept is $(1.002,1)$.
Based from the domain, the vertical asymptote is:
$$x=1$$
The sketch of the graph is as shown.
