# exponential function

exponential function
1. the function y = ex.
2. any function of the form y = bax, where a and b are positive constants.
3. any function in which a variable appears as an exponent and may also appear as a base, as y = x2x.
[1890-95]

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In mathematics, a function in which a constant base is raised to a variable power.

Exponential functions are used to model changes in population size, in the spread of diseases, and in the growth of investments. They can also accurately predict types of decline typified by radioactive decay (see half-life). The essence of exponential growth, and a characteristic of all exponential growth functions, is that they double in size over regular intervals. The most important exponential function is ex, the inverse of the natural logarithmic function (see logarithm).

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in mathematics, a relation of the form y = ax, with the independent variable x ranging over the entire real number line as the exponent of a positive number a. Probably the most important of the exponential functions is y = ex, sometimes written y = exp (x), in which e (2.7182818…) is the base of the natural system of logarithms (logarithm) (ln). By definition x is a logarithm, and there is thus a logarithmic function that is the inverse of the exponential function (see figure—>). Specifically, if y = ex, then x = ln y. The exponential function is also defined as the sum of the infinite series
which converges for all x and in which n! is a product of the first n positive integers. Thus in particular, the constant

The exponential functions are examples of nonalgebraic, or transcendental, functions—i.e., functions that cannot be represented as the product, sum, and difference of variables raised to some nonnegative integer power. Other common transcendental functions are the logarithmic functions and the trigonometric functions. Exponential functions frequently arise and quantitatively describe a number of phenomena in physics, such as radioactive decay (analysis), in which the rate of change in a process or substance depends directly on its current value.

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Universalium. 2010.

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