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

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

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

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

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