In mathematics, the exponential function is the function ex, where e is the number (approximately 2.718281828) such that the function ex equals its own derivative. The exponential function is used to model phenomena when a constant change in the independent variable gives the same proportional change (increase or decrease) in the dependent variable. The function is often written as exp(x), especially when the input is an expression too complex to be written as an exponent.
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 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 older sources refer to the exponential function as the anti-logarithm.
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.
Part of a series of articles on
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
Full article ▸