In mathematics, an

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

in which the input variable

*x* occurs as an exponent. A function of the form

$f(x)=b^{x+c}$, where

$c$ is a constant, is also considered an exponential function and can be rewritten as

$f(x)=ab^{x}$, with

$a=b^{c}$.

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In mathematics, an **exponential function** is a function of the form

in which the input variable *x* occurs as an exponent. A function of the form $f(x)=b^{x+c}$, where $c$ is a constant, is also considered an exponential function and can be rewritten as $f(x)=ab^{x}$, with $a=b^{c}$.

As functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function (i.e., its derivative) is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base $b$:

The constant *e* ≈ 2.71828... is the unique base for which the constant of proportionality is 1, so that the function's derivative is itself:

Since changing the base of the exponential function merely results in the appearance of an additional constant factor, it is computationally convenient to reduce the study of exponential functions in mathematical analysis to the study of this particular function, conventionally called the "natural exponential function", or simply, "the exponential function" and denoted by

The exponential function satisfies the fundamental multiplicative identity

(In fact, this identity extends to complex-valued exponents.) It can be shown that every continuous, nonzero solution of the functional equation $f(x+y)=f(x)f(y)$ is an exponential function, $f:\mathbb {R} \to \mathbb {R} ,\ x\mapsto b^{x}$, with $b>0$.

The argument of the exponential function can be any real or complex number or even an entirely different kind of mathematical object (*e.g.*, a matrix).

Its ubiquitous occurrence in pure and applied mathematics has led mathematician W. Rudin to opine that the exponential function is "the most important function in mathematics". In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease) in the dependent variable. Such a situation occurs widely in the natural and social sciences; thus, the exponential function also appears in a variety of contexts within physics, chemistry, engineering, mathematical biology, and economics.

The graph of $y=e^{x}$ is upward-sloping, and increases faster as $x$ increases. The graph always lies above the $x$-axis but can get arbitrarily close to it for negative $x$; thus, the $x$-axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its $y$-coordinate at that point, as implied by its derivative function (*see above*). Its inverse function is the natural logarithm, denoted $\log$, $\ln$, or $\log _{e}$; because of this, some old texts refer to the exponential function as the **antilogarithm**.

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