Why is this limit true in the distribution sense?

Question

$$ \lim_{\epsilon \rightarrow 0^+} \log(x+i\epsilon)=\log|x| +i \pi\theta(x) $$ Our professor told us, that the above limit in the distribution sense can be written in this way. Here we deal with the main branch of the complex logarithm. Since negative values are allowed, we can plug in -1. $e^{i\pi} = -1$ but $i\pi=\log(-1)\neq \log(1)+ +i \pi\theta(-1) =0$ by definition of the Heavy Side -function. What part am I missing? Thanks guys!