## The Additive Group $\R$ is Isomorphic to the Multiplicative Group $\R^{+}$ by Exponent Function

## Problem 322

Let $\R=(\R, +)$ be the additive group of real numbers and let $\R^{\times}=(\R\setminus\{0\}, \cdot)$ be the multiplicative group of real numbers.

**(a)** Prove that the map $\exp:\R \to \R^{\times}$ defined by

\[\exp(x)=e^x\]
is an injective group homomorphism.

**(b)** Prove that the additive group $\R$ is isomorphic to the multiplicative group

\[\R^{+}=\{x \in \R \mid x > 0\}.\]