Question: Prove that the group $G$ consisting of the set $\mathbb{R}\setminus\{-1\}$ with multiplication defined by $a\sim b=a+b+ab$ is isomorphic to the multiplicative group of nonzero real numbers, $(\mathbb{R}\setminus\{0\}, \times)$.

I'm trying to find a function that equates the two groups. Let's call the isomorphism Y. I've tricked around with $Y(a\sim b)=(ab+a+b)-(a+b)$, but this is taking in two input variables, so when it comes to trying to show $Y(a\sim b)=Y(a)(b)$, the function loses all of its meaning. So then I tried $Y(a)=-a$, but then $Y(a\sim b)=-a-b-ab \neq Y(a)y(b) = ab$.

A little help here? It's probably something obvious, but I'm having difficulties at forming a legitimate map.