Take a disk of radius \(R\) with a branch cut on the negative real axis; what does \(\log z\) map this onto? Where is the origin mapped onto?
The source does not state the center of the disk. The question about the image of the origin suggests the disk \(\{z : |z| < R\}\) centered at \(0\), with the cut along \((-R, 0]\); the source does not say so, and \(\log z\) is not defined at the branch point \(z = 0\).