On page 97 of The Road To Reality by Penrose he has Fig 5.9:

........+...........................................
....................+...............................
.....................*...+..........*.............
..............*................................*...
...........*..................+....................
.........*..........................................
........*...........................................
.......*............................+..............
....................................................
....................................................
......*............................................
......................................+............
......*..........*.*..............................
...............*......*...........+...............
.......*........*+*..*.......+..................
.....................................................
.......................+.*.+.......................
...........*...........*............................
..............*.....*..............................
.....................................................
So that + and * both start out from the origin, anti and clockwise
respectively


With the figure it says:


"The different values of W^z ( = e^(zlogw) ). Any integer multiple of
2ipi can be added to logw, which multiplies or divides w^z by e^(z
2ipi) an integer number of times. In the general case, these are
represented in the complex plane as the intersection of two
equiangular spirals (each making a constant angle with straight lines
through the origin).

Exercise 5.9 asks to show this.

How?

I did that one. I don't remember the details, but it involved solving the differential equation that expresses that it makes a constant angle with a line from the origin. It was an exponential spiral of some sort.
Laura

I don't think it's correct to prove it using the solution of a differential equation because differentiation isn't covered until chapter 6.