From powers to the exponential function

In Year 1 we defined $a^n$ for natural $n$ (product of $n$ copies of $a$), and in Year 2 we extended it to negative integer exponents ($a^{-n}=1/a^n$), to rational ones ($a^{m/n}=\sqrt[n]{a^m}$) and we stated the laws of powers:

$a^x \cdot a^y = a^{x+y},\quad \frac{a^x}{a^y} = a^{x-y},\quad (a^x)^y = a^{xy},\quad (ab)^x = a^x b^x.$

These laws had been proved only for rational exponents. In Year 3 we take the next step: we accept as a fact (provable with the tools of Analysis) that $a^x$ is well defined for every $x\in\mathbb{R}$ — not only rational — and that the same laws continue to hold. The resulting object is a function in the full sense.

Definition — Exponential function

Given a real number $a > 0$, $a \ne 1$, the exponential function with base $a$ is $\exp_a:\ \mathbb{R}\to(0;+\infty),\qquad \exp_a(x) = a^x.$

The domain is the whole of $\mathbb{R}$: we can raise the base to any real exponent. The codomain, on the other hand, is the set of positive reals: a power with a positive base is never zero nor negative.

Links

Topics: Exponential function
Concepts: Base · Power · Laws of powers
Functions: Exponential function