how to prove homomorphism of groups with real numbers

Question

I have this,
Let $\phi: G_1 → G_2$ defined by $\phi(x) = 2^x$ , where $G_1$ is a group of real numbers under addition and $G_2$ is a group of non-zero real numbers. How to show that $\phi$ is a homomorphism?

I know I need to prove this, $\phi(xy) = \phi(x)\phi(y)$

Answer 1

The group operation on $G_1$ is addition, but the group operation on $G_2$ is multiplication. So what you have to show (among other things) is that $$\phi(x+y) = \phi(x) \times \phi(y)$$