Why Is the group ${\mathbb R}$ with the operation addition and group ${\mathbb R^{pos}}$ with the operation multiplication ISOMORPHIC

Question

The function that makes this two groups Isomorphic was f(x) = $e^x$. I know Isomorphic groups are bijective. So my concern is if $\mathbb R^{pos}$ refers to all real positive number, how will negative element in $\mathbb R$ be to the $\mathbb R^{pos}$

Answer 1

The domain of $f$ is taken as $\mathbb R$ and the range $\mathbb R^{pos}$. This map is a bijection and a homomorphism [with $f^{-1}(y)=\ln y$].

Answer 2

Exactly as you have said, $-1$ is mapped to $e^{-1}$ which is a positive number.