# Logarithm of a complex number as intersections of two logarithmic spirals

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?

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It would have been nice to post a reference. This is regarding Section 5.4 and Figure 5.9. I find his description unclear, too. The spirals in the figure are $w^z$ for various choices of $\log w$, but I don't get the emphasis on the intersections. – Ross Millikan Jan 23 '11 at 16:08
One of the spirals should be $\exp(z \operatorname{Log} w) \cdot \exp(2\pi i z t)$ for $t$ real, which gives the values of $w^z$ when $t=n$ is an integer, but I don't see immediately what the other spiral is. – Hans Lundmark Jan 23 '11 at 16:15
Thank you for the responses. I must rather shamefacedly admit that I had already looked at an answer to this on the 'Road to Reality' Internet Forum at roadtoreality.info/viewtopic.php?f=19&t=100, but the caveats expressed by the site owner made me doubt it and look for a simpler answer. Now I have studied that reference carefully I can see that it is correct and complete: but no way could I have worked it out on my own. – Harry Weston Jan 24 '11 at 15:09