Answer
Please see below.
Work Step by Step
The graph of $h(x)= \ln (-x)$ (the red graph) is the graph of $f(x)=\ln x$ (the blue graph) reflected about the $y$-axis.
Looking at the graph, we observe that the vertical asymptote is $x=0$.
We also find that the domain and the range of the function $h(x)$ are$$D_h=(- \infty, 0 ) \quad \text{ and } \quad R_h= (- \infty, \infty ),$$respectively.