Answer
Domain :$\qquad(-5,+\infty)$
Range :$\qquad(-\infty,+\infty)$
Vertical asymptote: $\qquad x=-5$
Work Step by Step
We start with the graph of $f_{1}(x)=\ln(x)$ (dashed blue).
Since $h(x)=\ln(x+5)=f(x+5)$
its graph (red) is obtained by
shifting $f_{1}(x)$ to the left by $5$ units.
Domain :$\qquad(-5,+\infty)$
Range :$\qquad(-\infty,+\infty)$
Vertical asymptote: $\qquad x=-5$