Here is the picture of log(z). Note where the log is equal to 0 (black), and where it becomes large in modulus (white). What is happening along the negative reals? Why does the color jump? Ans: the machine is using the principal branch of the log, which jumps from the negative imaginary axis (purple in the standard plane) to the positive imaginary axis (yellow-green in the standard plane). What is happening between 0 and 1? Ans: the color is cyan or light blue, because the log is negative there (cyan is the color of the negative reals in the standard plane).
What will other "branches" of the logarithm look like?