Editing The Road to Reality Study Notes (section)

=== 5.2 The idea of the complex logarithm ===

Armed with both the cartesian and polar representations of complex numbers, it is now possible to show that the multiplication of two complex numbers leads to adding their arguments and multiplying the moduli.  This, for the moduli, converts multiplication into addition.

This idea is fundamental in the use of logarithms.  We first start with the expression <math>b^{m+n} = b^m \times b^n</math>, which represents the idea of [https://en.wikipedia.org/wiki/Exponentiation exponentiation]. This is easy to grasp for <math>m</math> and <math>n</math> being positive integers, as each side just represents <math>m+n</math> instances of the number <math>b</math>, all multiplied together.  If <math>b</math> is positive, this law is then showed to hold for exponents that are positive integers, values of 0, negative, and fractions.  If <math>b</math> is negative, we require further expansion into the complex plane. 

We would need a definition of <math>b^p</math> for all complex numbers <math>p,q,b</math> such that <math>b^{p+q} = b^p \times b^q</math>.  If we then define the [https://en.wikipedia.org/wiki/Logarithm logarithm to the base b] as the inverse of the function <math>f(z) = b^z</math> such that <math>z=log_bw</math> for <math>w=b^z</math> then we should expect <math>z=log_b(p \times q) = log_bp + log_bq</math>.  This would then convert multiplication into addition and allow for exponentiation in the complex plane.