Hint : the map maps the upper half-plane to the unit disc , and the map maps the slit plane to the upper half-plane, provided you pick the appropriate branch of the function.
Find a conformal map of the slit plane onto the open unit disk satisfying . What are the images of circles centered at 0 under the map? Sketch them.
The back of the book says that works. I think for the second part of the question I can consider three points on the circles to find the images. However, I don't see how they found this map. Thanks.