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Definition

exponential function

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

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