An exponential function is a mathematical function of the following form:

*f* ( *x* ) = *a ^{x}*

where *x* is a variable, and *a* is a constant called the base of the function. The most commonly encountered exponential-function base is the transcendental number *e* , which is equal to approximately 2.71828. Thus, the above expression becomes:

*f* ( *x* ) = *e ^{x}*

When the exponent in this function increases by 1, the value of the function increases by a factor of *e* . When the exponent decreases by 1, the value of the function decreases by this same factor (it is divided by *e* ).

In electronics and experimental science, base-10 exponential functions are encountered. The general form is:

*f* ( *x* ) = 10 ^{x}

When the exponent increases by 1, the value of the base-10 function increases by a factor of 10; when the exponent decreases by 1, the value of the function becomes 1/10 as great. A change of this extent is called one order of magnitude.

For a given, constant base such as *e* or 10, the exponential function "undoes" the logarithm function, and the logarithm undoes the exponential. Thus, these functions are inverses of each other. For example, if the base is 10 and *x* = 3:

log (10 ^{x} ) = log (10 ^{3} ) = log 1000 = 3

If the base is 10 and *x* = 1000:

10 ^{(log x)} = 10 ^{(log 1000)} = 10 ^{3} = 1000

*This was last updated in*September 2005

*Posted by:*Margaret Rouse

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