existence of a unique function with property of equivalent to the exponential

Question

Suppose that we wanted to prove that there exists a unique function $Exp:\mathbb{R}\rightarrow\mathbb{R}$ with the property that $Exp(q)=exp(q)$ for $q\in\mathbb{Q}$ where $exp(q)$ is the exponential function. How would we prove this? I know that the exponential function is uniformly convergent on the $\mathbb{R}$. Would it be sufficient to define $Exp$ as the the exponential form and show that for all $q\in\mathbb{Q}$ we have that $Exp(q)=exp(q)$? I can not think of any other way to prove it.