How do you find the domain and range of g(x) = ln(x + 4)?

The domain is $x > - 4$ .
g(x) is a really the function $\ln \left(x\right)$ shifted 4 units to the left. The domain for the function $\ln \left(x\right)$ is $x > 0$. Note that the function $\ln \left(x\right)$ does not exist where $x = 0$, that is, at $x = 0$ there is a vertical asymptote of the function.
When shifting the function 4 units left, the asymptote shifts from $x = 0$ to $x = - 4$ . As a result, the function, $g \left(x\right)$, is defined where $x > - 4$ - and this is the domain.