Answer
The domain: $x\lt0$.
The $x$-intercept: $\left(-\frac{1}{2},0\right)$.
The vertical asymptote: $x=0$.
See graph
Work Step by Step
$$f(x)=\log (-2x)$$
The function is defined when:
$$-2x\gt0$$ $$-2x(-\frac{1}{2})\lt0(-\frac{1}{2})$$ $$x\lt0$$
Thus, the domain is $x\lt0$.
Finding the $x$-intercept, set $f(x)=0$:
$$0=\log (-2x)$$ $$\log (-2x)=0$$
Rewriting in exponential form:
$$-2x=10^{0}$$ $$-2x=1$$ $$x=-\frac{1}{2}$$
Thus, the $x$-intercept is $\left(-\frac{1}{2},0\right)$.
Based from the domain, the vertical asymptote is:
$$x=0$$
The sketch of the graph is as shown.
