I have to map $z=e^w$ to the $w-plane$; where, $f(z)=ln(z)$
Work done so far:
Since $ln(z)=ln(|z|)+i(\theta+2k\pi)$ where $k\in \mathbb Z$ and $-\pi\lt\theta \lt \pi$
I get: $ln(z)=ln(e)+i(0+2k\pi)$, since $|z|=e$.
However, I am not sure how to plot $ln(z)$. I know $u$ is static; that is, in $w=u+vi$ $u=1$ will not change. In summary, how should I plot points for a $ln(z)$ so that I can estimate a rough sketch of the transformation.
Any help will be appreciated, thanks in advance!