Math Problem Solver

Let $a$ be a positive real number. Then, since $a^{x+y}=a^xa^y$, the mapping $a\mapsto a^x$ is a homomorphism from the additive group $(\mathbb{Q},+)$ to the multiplicative group $(\mathbb{R}^+,\cdot)$ of all positive real numbers. This will in fact be a momomorphism, if $a\ne 1$.

If $a\ne 1$, is it possible to extend this monomorphism to an isomorphism from $\mathbb{R}$ to $\mathbb{R}^+$, WITHOUT it being the mapping $a\mapsto a^x$ for all $x\in\mathbb{R}$? Note: Nothing is said that the mapping must be continuous.