Are the groups$ (\mathbb{R}^\times, \cdot)$ and $((\mathbb{R} \times \mathbb{Z}_2), +)$ isomorphic

Question

I have attempted to find a bijection from $\mathbb{R}^\times$ to $(\mathbb{R}\times \mathbb{Z}_2)$ but can't seem to get anything to work. Any suggestions would be greatly appreciated.

Answer 1

$$e^x\mapsto (x,0)$$ $$-e^x\mapsto (x,1)$$ Gives the isomorphism.