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