Well-known isomorphism

Question

Question: What is the well-known isomorphism from the multiplicative group of positive reals to the additive group of all real numbers?
There is a well-known isomorphism out there, I cannot find. If anyone knows, I would appreciate it

Answer 1

You are thinking of the isomorphism $(\mathbb{R},+) \to (\mathbb{R}^+,\times)$ given by $x \mapsto e^x$ with inverse $x \mapsto \log{x}$. Check for yourself that this is a homomorphism and that it's bijective.