Understanding the logarithm is crucial to understanding many of the subtleties of functions in the complex plane. f(z) offers many tools for exploring the logarithm.
These are selected frames from an animation in which the circles in the left figure (the domain) move through the singularity at the origin. The right figure is the logarithm of the left.
Note that when circles avoid the origin the images are simple. As they cross the origin, the images "open up" and approach the familiar vertical and horizontal lines of the last frame.
f(z) can continuously follow a branch of the logarithm - even when drawing freehand.
