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The exponential function is a mathematical function denoted by $f\left(x\right)=\exp \left(x\right)$ or $e^x$ (where the argument x is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the operation of taking powers of a number (repeated multiplication), but various modern definitions allow it to be rigorously extended to all real arguments $x$, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to consider the exponential function to be "the most important function in mathematics".

The function $f\left(x\right)=b^x$ for any positive real number $b$ (called the base) is also known as a (general) exponential function, and satisfies the exponentiation identity:$b^{x+y}=b^x{b}^y\text{ for all }x,y\in R.$This implies $b^n=b\times \cdots \times b$ (with $n$ factors) for positive integers $n$, where $b=b^1$, relating exponential functions to the elementary notion of exponentiation. The natural base $e=\exp \left(1\right)=2.71828\dots$ is a ubiquitous mathematical constant called Euler's number. To distinguish it, $\exp \left(x\right)=e^x$ is called the exponential function or the natural exponential function: it is the unique real-valued function of a real variable whose derivative is itself and whose value at 0 is 1:

${\exp }^{\prime }\left(x\right)=\exp \left(x\right)$ for all $x\in R$, and $\exp \left(0\right)=1.$

The relation $b^x=e^{x\ln b}$ for $b>0$ and real or complex $x$ allows general exponential functions to be expressed in terms of the natural exponential.

More generally, especially in applied settings, any function $f:R\to R$ defined by

$f\left(x\right)=a{e}^{cx}=a{b}^{kx},\text{ with }c=k\ln b,c\ne 0,a,b>0$

is also known as an exponential function: it solves the initial value problem $f^{\prime }=af,f\left(0\right)=a$, meaning its rate of change at each point is proportional to the value of the function at that point. This behavior models diverse phenomena in the biological, physical, and social sciences, for example the unconstrained growth of a self-reproducing population, the decay of a radioactive element, the compound interest accruing on a financial fund, or the self-sustaining improvement of computer design.

The exponential function can also be defined as a power series, which is readily applied to real, complex, and even matrix arguments. The complex exponential function $\exp :C\to C$ takes on all complex values except 0 and is closely related to the trigonometric functions by Euler's formula:

$e^{x+iy}=e^x\cos \left(y\right)+i{e}^x\sin \left(y\right).$

Motivated by its more abstract properties and characterizations, the exponential function can be generalized to much larger contexts such as square matrices and Lie groups. Even further, the differential equation definition can be generalized to a Riemannian manifold.

The exponential function for real numbers is a bijection from $R$ to the interval $(0,\infty )$. Its inverse function is the natural logarithm, denoted $\ln$, $\log$, or ${\log }_e$. Some old texts refer to the exponential function as the antilogarithm.