TITLE: Why doesn't $e^x$ have an inverse in the complex plane?
QUESTION [5 upvotes]: Why doesn't $e^x$ have an inverse in the complex plane? Can someone please clarify it?

REPLY [4 votes]: Here comes the overkill: by Great Picard's Theorem, any analytic function with an essential singularity at infinity takes every complex value, with at most one exception, an infinite number of times. $e^z$ clearly has an essential singularity at infinity.