Is there a name for a function such that $f=e^g$? Let $X$ be a topological space.
Let $f:X\rightarrow \mathbb{C}\setminus\{0\}$ be a continuous function.
Is there a terminology to call functions $f$ such that $f=e^g$ for some continuous map $g:X\rightarrow \mathbb{C}$?
Here is an example such function is used:

Theorem
Let $X$ be a compact metric space.
Let $f,g:X\rightarrow \mathbb{C}\setminus\{0\}$ be continuous functions.
Then, $f,g$ are homotopic iff there exists a continuous function $h:X\rightarrow \mathbb{C}$ such that $f/g=e^h$.

What are these functions $f$ called such that $f=e^g$?
 A: There is a name - for $X=S^1$ these are called paths with winding number zero, and the theorem statement is just a generalization of this to arbitrary compact sets. Since the fundamental group of $\Bbb C^\times$ is nontrivial, there are non-nullhomotopic maps that wrap around the origin. The exponential map is a covering map from $\Bbb C\to\Bbb C^\times$, so the technique of demanding the existence of a $g$ such that $f=e^g$ is basically lifting the path from the "twisted" space $\Bbb C^\times$ onto its "untwisted" covering space $\Bbb C$, which is nullhomotopic. Thus we can do our homotopy work in this space and then project back into $\Bbb C^\times$ to get a path homotopy. Some basic facts about covering maps allow us to run this process backwards in the sense that any homotopy can be lifted to a homotopy in the simpler space, so this becomes an if and only if. It's the same idea as the covering of a circle by a line - if a path on the circle is the image of a closed path on the line, then the path on the circle is nullhomotopic, while if the path on the circle winds once around, then the path on the line will not be closed (i.e. continuous) and there is no homotopy.
