## image, principal value mapping

Sketch an image under $w = \text{Log} (z)$ of

i) the line $y=x$
ii) the line $x=e$

I have not done many problems with mappings under the principal value mapping of the logarithm. I know that

$\text{Log} (z) := \ln r + i \theta = \ln | z | + i \text{Arg} (z)$.

However, I do not see what these images would look like. I need a few pointers here on what to do.