How to go from a sum to a product and a product to a sum? I have read here (third post down) that exponentials turn sums into products and logarithms turn products into sums. Can someone please further explain this?

Consider the set $\mathbb R$ of all real numbers together with the operation of addition and consider the set $\mathbb R_+$ of all positive real numbers with the operation of multiplication. Both of these are examples of groups. The properties that $e^{a+b}=e^a\cdot e^b$ and $\ln(ab)=\ln(a)+\ln(b)$, show that $\exp:(\mathbb R,+)\to (\mathbb R_+,\cdot)$ is a group isomorphism. Its inverse is $\ln$. 
