Exponential Function (eˣ) ○◂ Entries

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Exponential Function (eˣ) ○◂｜Definition｜1st｜20260521132350-00-⌔

Exponential function - Wikipedia

Exponential function

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In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted ⁠ $e^x$ ⁠ or ⁠ $\exp x$ ⁠; the latter is preferred when the argument ⁠ $x$ ⁠ is a complicated expression.¹² It is called exponential because its argument can be seen as an exponent to which a constant number e ≈ 2.718, the base, is raised. There are several other definitions of the exponential function, which are all equivalent although being of very different nature.

The exponential function converts sums to products: ⁠ $\exp (x+y)=\exp x\cdot \exp y$ ⁠. Its inverse function, the natural logarithm, ⁠ $\ln$ ⁠ or ⁠ $\log$ ⁠, converts products to sums: ⁠ $\ln (x\cdot y)=\ln x+\ln y$ ⁠.

The exponential function is occasionally called the natural exponential function, matching the name natural logarithm, for distinguishing it from some other functions that are also commonly called exponential functions. These functions include the functions of the form ⁠ $f(x)=b^x$ ⁠, which is exponentiation with a fixed base ⁠ $b$ ⁠. More generally, and especially in applications, functions of the general form ⁠ $f(x)=a{b}^x$ ⁠ are also called exponential functions. They grow or decay exponentially in that the rate that ⁠ $f(x)$ ⁠ changes when ⁠ $x$ ⁠ is increased is proportional to the current value of ⁠ $f(x)$ ⁠.

The exponential function can be generalized to accept complex numbers as arguments. This reveals relations between multiplication of complex numbers, rotations in the complex plane, and trigonometry. Euler’s formula ⁠ $e^{i\theta }=\cos \theta +i\sin \theta$ ⁠ expresses and summarizes these relations.

The exponential function can be even further generalized to accept other types of arguments, such as matrices and elements of Lie algebras.

Printed 2026-06-28.

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Footnotes

1. “Reviews of Modern Physics Style Guide” (PDF). XVI.B.1(d): American Physical Society. p. 18. Retrieved 2025-12-30. Which form to use, ⁠$e$⁠ or ⁠$\exp$⁠, is determined by the number of characters and the complexity of the argument. The ⁠$e$⁠ form is appropriate when the argument is short and simple, i.e., $e^{i\mathbf{k}\cdot \mathbf{r}}$, whereas ⁠$\exp$⁠ should be used if the argument is more complicated. ↩

2. T. W. Chaundy; P. R. Barrett; Charles Batey (1954). The Printing of Mathematics. Oxford University Press. p. 31. ↩

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