An exponential function places the variable in the exponent:

$$f(x)=a^x,$$

with $a>0$ and $a\ne1$.

For integer inputs, the exponent has its familiar meaning. For example,

$$2^3=8,\qquad 2^{-2}=\frac14.$$

The exponential function extends this pattern continuously to real inputs.

Growth and decay

If $a>1$, the function increases and represents exponential growth. If $0<a<1$, it decreases and represents exponential decay.

A scaled exponential

$$f(x)=Ca^x$$

has initial value $f(0)=C$. Multiplying the input by one more unit multiplies the output by the constant factor $a$:

$$f(x+1)=a f(x).$$

This constant multiplicative change is what distinguishes exponential behavior from linear growth.

Graph and range

For $a>0$, every value $a^x$ is positive. The graph passes through $(0,1)$ and approaches the horizontal axis as $x$ extends in one direction, but never reaches zero.

The number $e$

A particularly important base is

$$e\approx2.71828.$$

The function $e^x$ appears naturally throughout calculus, differential equations, probability and models of continuous growth and decay.