Let a be a positive number not equal to 1. The exponential function to base a is the function f such that f(x) = a^x for all x in ℝ. The graphs y = 2^x and $y={\left(\frac{1}{2}\right)}^x$ illustrate the essential difference between the cases when a>1 and a<1. See also exponential decay, exponential growth.

Graph showing exponential growth

Graph showing exponential decay

As x may not be an integer (or rational number even) a^x may not be defined as ‘a multiplied by itself x times’ and instead is defined by

This definition still satisfies rules (i)–(vii) of the rules for indices (see index). Further, by the chain rule,

$\frac{\text{d}}{\text{d}x}\left(a^x\right)=a^x\text{ln}a.$