There exist beautiful numerical approximation for calculation of the function $$f(z) = \log(1+\exp(z)).$$ In case if $z$ is real, the following can be used $$f(z) = \begin{cases} z & z \gg 1 \\ e^{z} & z \ll 1 \\\log(1 + e^z) & \text{otherwise} \end{cases}$$
But what about the case when $z$ is complex? What should be the conditions for using the same kind of approximation?