# exponential

The graph of *y = e ^{x}* and its inverse,

*y*= ln

*x*.

Who has not be amazed to learn that the function y = e^{x}, like a phoenix rising
again from its own ashes, is its own derivative? |

– Francois le Lionnais |

An exponential is anything that grows at a rate proportional to its size. An

**exponential function**is a function of the form

*y = a*, where

^{x}*a*is an arbitrary constant greater than zero and

*x*is variable between -∞ and +∞.

An exponential function has the property:

*a*^{m} .* a*^{n} = *a*^{m + n}

If *a = e* we have the important exponential function *y = e ^{x}*, where

*e*is about 2.712... The inverse of this function is

*y*= ln

*x*. The exponential function to base

*a*can be written as

*f*(

*x*) =

*a*

^{x}.

It was proved by Leibniz that exponential functions are transcendental.