Why homomorph and not isomorph?

Question

Why are the groups $\mathbb{R},+$ and $\mathbb{R}_0^+,*$ homomorph, their mapping function being $ f: x \rightarrow e^x $? Why is this not an isomorphism?

Answer 1

These groups are isomorphic, in fact $f$ is an isomorphism with inverse map $x\mapsto \log(x)$, i.e., with $\exp\circ \log=\log\circ \exp=id$.