TITLE: Logarithm of a complex number as intersections of two logarithmic spirals
QUESTION [5 upvotes]: In Penrose's book "The Road to Reality" page 97 figure 5.9 he shows the values of the complex logarithm in a diagram as the intersection of two logarithmic spirals. Can you please explain how these particular spirals are derived from the definition of the logarithm?

REPLY [2 votes]: If $z = x+iy$ and $w = \exp(x'+iy')$, then the set of all powers $w^z$ is given by
$$ \{ w^z \} = \{ \exp(z\log w) \} = \{\exp(xx'-yy'-2\pi ky +i(xy'+x'y+2\pi k x) : k \in \Bbb{Z} \}$$
You can easily check that this set lies on the logarithmic (equiangular) spiral given in polar coordinates $(r,\theta)$ by
$$ r(t) = \exp(\alpha_m t + \beta_m), \quad \theta(t) = t $$
where
$$ \alpha_m = \frac{y}{m-x},\quad \beta_m = xx' - yy' - \alpha_m(xy'+x'y) $$
and $m \in \Bbb{Z}$ is an arbitrary integer $\neq x$. Moreover, you can also easily check that the intersection of any two such spirals for integer $m,n$ with $|m-n| = 1$ is precisely the above set of all complex powers $w^z$. The surprising thing is that this set in itself can suggest a shape (see this picture) that is more complicated than a logarithmic spiral, despite (or because of) the fact that it lies on infinitely many of them.