Multi-valued logarithmic function

Question

I'm reading some notes for an electrical engineering class and came to the following: "...$2^j$ can represent a countably infinite number of real numbers. These examples are related to the fact that if we define w = log(z) to mean that $z=e^w$, then for $z \neq 0$, this logarithm function log(z) is multi-valued."

In here, I was wondering what they mean by multi-valued. There is no further description of this function on the text. Are they referring to the complex logarithmic function? Is the non-complex logarithmic function multi-valued too?

Answer 1

The real logarithm function, $\log:\mathbb{R}^+\to\mathbb{R}$ is certainly not multi-valued, as the real exponential function is bijective from $\mathbb{R}\to \mathbb{R}^+$

The complex logarithm has some multi-valued issues because the complex exponential has periodic behavior. That is, $e^{x+2\pi i} = e^x$, so the logarithm needs to reflect this complication somehow.

Answer 2

For the complex logarithm, consider the fact that $i^4=1.$ Then $$\ln(z) =\ln(z\cdot i^{4k})=\ln(z) +4k\ln(i)=\ln(z)+2k\pi i. $$