The conditions on the base

In the definition of the exponential function the base $a$ is not just any number: it must be positive and different from $1$. Let us see why neither of these conditions can be dropped.

Warning — The conditions on the base

The conditions $a>0$ and $a\ne 1$ are essential:

• If $a = 0$: $0^x$ is not defined for $x\le 0$.
• If $a < 0$: $a^{1/2}=\sqrt{a}$ does not exist in $\mathbb{R}$, so the function would not be defined on the whole of $\mathbb{R}$.
• If $a = 1$: $1^x = 1$ for every $x$, that is a constant function, of no interest whatever.

Only by choosing $a>0$ and $a\ne 1$ do we obtain a function defined on the whole real line and genuinely “interesting”, that is strictly increasing or strictly decreasing.

Links

Topics: Exponential function
Concepts: Base
Functions: Exponential function