Exponential Function

In mathematics, the exponential function is the function ex, where e is the number (approximately 2.718281828) such that the function ex is its own derivative. The exponential function is used to 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. The function is often written as exp(x), especially when it is impractical to write the independent variable as a superscript. The exponential function is widely used in physics, chemistry and mathematics.

Exponential Function
Representation
Inverse
Derivative
Indefinite Integral

The graph of y = ex 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. The inverse function is the natural logarithm ln(x); because of this, some old texts refer to the exponential function as the antilogarithm.

Sometimes the term exponential function is used more generally for functions of the form cbx, where the base b is any positive real number, not necessarily e. See exponential growth for this usage.

In general, the variable x can be any real or complex number, or even an entirely different kind of mathematical object; see the formal definition below.


The mathematical constant e

Natural logarithm · Exponential function

Applications in: compound interest · Euler's identity & Euler's formula · half-lives & exponential growth/decay

Defining e: proof that e is irrational · representations of e · Lindemann–Weierstrass theorem

People John Napier · Leonhard Euler

Schanuel's conjecture

Read more about Exponential Function:  Overview, Formal Definition, Derivatives and Differential Equations, Continued Fractions For ex, Complex Plane, Matrices and Banach Algebras, On Lie Algebras, Double Exponential Function, Similar Properties of e and The Function ez

Famous quotes containing the word function:

    As a medium of exchange,... worrying regulates intimacy, and it is often an appropriate response to ordinary demands that begin to feel excessive. But from a modernized Freudian view, worrying—as a reflex response to demand—never puts the self or the objects of its interest into question, and that is precisely its function in psychic life. It domesticates self-doubt.
    Adam Phillips, British child psychoanalyst. “Worrying and Its Discontents,” in On Kissing, Tickling, and Being Bored, p. 58, Harvard University Press (1993)