# exponent

An exponent is a number that gives the power to which
a base is raised. For example, in 3^{2} the base is 3 and the exponent is 2.

## Exponential equation

An equation of the form *a ^{x}* =

*b*where

*a*and

*b*are numbers. For example, if 3

^{x}= 81 we can solve for

*x*by restating 81 = 9 × 9 = 3 × 3 × 3 × 3, and so

*x*= 4. More difficult problems are solved using logarithms: e.g., if 4

^{x}= 15, then

*x*log

_{e}4 = log

_{e}15. Therefore 1.386

*x*= 2.7081 and

*x*= 1.9535 (to 4 decimal places).

## Exponential function

A function of the form *f(x)* = *a ^{x}* where

*x*is positive and does not equal 1. In terms of differential calculus,

*f '(x)*=

*a*log

^{x}_{e}

*a*

*f(x)*=

*a*.

^{x}

## Exponential growth

Since the value of *a ^{x}* increases considerably with increases
in

*x*, the term exponential growth is used loosely in statistics to refer to the very rapid growth in number of a quantity over a period of time. More accurately, it refers to an increase which, if plotted against units of time, would approximate closely to an exponential curve (the plotting of an exponential function).