In Roger Penrose's book Road to Reality - Chapter 5 - he goes to great lengths to arrive at the standard polar expression for a complex number $w = r e^{i \theta}$ via a discussion of complex logarithms, see page 94-95 of the following link.
The key step, on page 94, is to look for an inverse to $w = e^z$ to define $\log(w) = z$. Penrose denotes the radius and argument of $w$ as $[r,\theta]$ and then just states that the logarithm of $w$ - i.e. $z$ - is:
$\log(w) = z = \log(r) + i \theta$.
Penrose then uses the above equation $\log(z) = \log(r) + i \theta$ to 'arrive at the result' that $z = \exp(w) = \exp(\log r + i \theta) = r \exp(i\theta).$
My question is: is it internally consistent / possible to arrive at the standard polar form of a complex number from the logarithm as Penrose does, or is this just circular? Put another way, does it make sense to claim the form of $\log(w) = z =\log(r) + i \theta$ above without reference to $z = r \exp(i\theta)$?
Couple more notes / comments
Penrose points out in the text below the expression for $\log(w) = z=\log(r) + i \theta$ that it is " remarkable [] that the imaginary part of $z$ is just the angle $\theta$ that is the argument of the complex number $w$". There is a footnote to this that points out that he doesn't claim to have proven that the $i \theta$ term could not be a real multiple of $i \theta$ since this "requires calculus" - but I don't know what this means.
My approach to getting to the logarithm expression would be to to observe that $e^{i\theta} = \cos\theta + i\sin\theta$ via power series expansions and then note that this is directly the definition of $w$ by basic trigonometry (real part of $w$ equals $r \cos\theta$ etc. etc). The expression for $\log(w)$ then follows automatically by properties of logarithms. But Penrose has very clearly gone via the logarithm argument to get at the polar form of a generic complex number, and he's a much smarter man than me, so I can't help but feel I'm missing something profound in his argument. What am I missing?