Chapter 3.5, Problem 13E

Consider the groups given in Exercise $12$. Find an isomorphism from the multiplicative group ${ℝ}^{+}$ of positive real numbers to the additive group $ℝ$ of real numbers.

Prove that the additive group $ℝ$ of real numbers is isomorphic to the multiplicative group ${ℝ}^{+}$ of positive real numbers. (Hint: Consider the mapping $\varnothing :ℝ\to {ℝ}^{+}$ defined by $\varnothing (x)={10}^x$ for all $x\in ℝ$.)