Suppose $b \in (0, \infty), b \ne 1$.

Definition: Exponential function base b for natural numbers. $exp_b : N \rightarrow R$ , the exponential function with the base $b$ is defined by

[ a recursive definition ] $exp_b(0) = 1, exp_b(n+1) = b \cdot exp_b(n)$, that is, $exp_b(n) = b^n$ for all $n \in N$.

Definition: Exponential function base b for integers. $exp_b : Z \rightarrow R$, the exponential function with the base $b$ is defined by $exp_b(n) = b^n$ for $n \in N$ and $exp_b(n)= \frac 1 {b^{-n}}$ for $n \in Z$ with $n <0$.
Definition: Exponential function base b for rational numbers. $exp_b : Q \rightarrow R$, the exponential function with the base $b$ is defined by $exp_b(\frac p q) = (\sqrt [q] b )^p = \sqrt [q] {b^p}$ for all $\frac p q \in Q, q > 0$.
Definition: Exponential function base b for real numbers. $exp_b : R \rightarrow R$, the exponential function with the base $b$ is defined by $exp_b(x) = \stackrel {lim }{_{\frac p q \rightarrow x}} exp_b(\frac p q)$ for all $x \in R$ with $\frac p q \in Q$.