Sketch an image under $\displaystyle w = \text{Log} (z)$ of the half disk $\displaystyle |z| < 1, \text{Re} (z) > 0$.

To this point, I do not have much experience with mappings under the principal value mapping of the logarithm. I know that

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

However, I do not see what this image would look like.