# Exponential function
The exponential function is the mathematical function \( \mathrm{exp}(x) = e^{x} \), where \( e \approx 2.718 \) is the base of the natural logarithm; it is its own derivative and turns addition of inputs into multiplication of outputs. In elementary contexts it is called the natural exponential function, and the more general family \( f(x) = a b^{cx+d} \), with positive base \( b \), reduces to it by a scaling of the variable.[1](https://mathworld.wolfram.com/ExponentialFunction.html)
| Key fact | Detail |
|---|---|
| Definition | \( \mathrm{exp}(x) = e^{x} \), with \( e = 2.718\ldots \) the base of the natural logarithm; equivalently the unique solution of \( df/dx = f \) with \( f(0) = 1 \)[1](https://mathworld.wolfram.com/ExponentialFunction.html) |
| Power series | \( \mathrm{exp}\,z = 1 + z/1! + z^{2}/2! + z^{3}/3! + \cdots \), convergent for every complex \( z \)[2](https://dlmf.nist.gov/4.2) |
| Functional equation | \( \mathrm{exp}(x+y) = \mathrm{exp}(x)\,\mathrm{exp}(y) \); also \( \mathrm{exp}(-z) = 1/\mathrm{exp}(z) \)[1](https://mathworld.wolfram.com/ExponentialFunction.html) • [2](https://dlmf.nist.gov/4.2) |
| Complex behavior | An entire function with no zeros and period \( 2\pi\mathrm{i} \); for \( z = x + \mathrm{i}y \), \( \mathrm{exp}\,z = e^{x}\cos y + \mathrm{i}e^{x}\sin y \)[2](https://dlmf.nist.gov/4.2) |
| Growth and decay | Models of the form \( y = y_{0}e^{kt} \) (\( k > 0 \)) or \( y = y_{0}e^{-kt} \) describe population growth, compound interest, radioactive decay, and Newton's law of cooling[3](https://openstax.org/books/calculus-volume-1/pages/6-8-exponential-growth-and-decay) |
| Floating-point limits | In the fdlibm double-precision implementation, \( \mathrm{exp}(x) \) overflows for \( x > 709.78\ldots \) and underflows for \( x < -745.13\ldots \)[4](https://code.googlesource.com/edge/openjdk/+/refs/heads/master/jdk/src/share/native/java/lang/fdlibm/src/e_exp.c) |
## Characterizations
Three definitions of the same function are in common use, each useful in a different setting. The differential characterization says that \( \mathrm{exp} \) is the unique solution of \( df/dx = f \) with \( f(0) = 1 \); in other words, the function is its own derivative, \( d(e^{x})/dx = e^{x} \).[1](https://mathworld.wolfram.com/ExponentialFunction.html) The series characterization expands it as
\[ \mathrm{exp}\,z = 1 + \frac{z}{1!} + \frac{z^{2}}{2!} + \frac{z^{3}}{3!} + \cdots, \]
which the NIST Digital Library of Mathematical Functions gives as the formal definition and which converges throughout the complex plane.[2](https://dlmf.nist.gov/4.2) • [5](https://encyclopediaofmath.org/wiki/Exponential_function) The limit characterization, \( \mathrm{exp}(z) = \lim_{n \to \infty}(1 + z/n)^{n} \), connects the function to repeated compounding.[1](https://mathworld.wolfram.com/ExponentialFunction.html)
The functional equation \( \mathrm{exp}(x+y) = \mathrm{exp}(x)\,\mathrm{exp}(y) \) is the property that makes the function an isomorphism from addition to multiplication: adding arguments multiplies values. From it follow \( \mathrm{exp}(-z) = 1/\mathrm{exp}(z) \) and \( |\mathrm{exp}\,z| = \mathrm{exp}(\Re z) \).[1](https://mathworld.wolfram.com/ExponentialFunction.html) • [2](https://dlmf.nist.gov/4.2)
## The complex exponential
Extended to a complex variable \( z = x + \mathrm{i}y \), the exponential function satisfies
\[ \mathrm{exp}\,z = e^{x}\cos y + \mathrm{i}\,e^{x}\sin y. \]
This formula ties the function to trigonometry and explains its complex-plane behavior: \( \mathrm{exp} \) is an entire function (analytic everywhere), it has no real or complex zeros, and it is periodic with period \( 2\pi\mathrm{i} \), since \( \mathrm{exp}(z + 2\pi\mathrm{i}) = \mathrm{exp}\,z \).[2](https://dlmf.nist.gov/4.2) The modulus of \( \mathrm{exp}\,z \) depends only on the real part \( x \), while the phase is \( \Im z + 2k\pi \) for integer \( k \), which is why the function is many-to-one on the complex plane.[2](https://dlmf.nist.gov/4.2)
The exponential and the logarithm are inverse functions where the logarithm is defined: for \( \zeta \ne 0 \), \( \mathrm{exp}\,z = \zeta \) exactly when \( z = \mathrm{Ln}\,\zeta \), the general (multivalued) logarithm. Powers with arbitrary bases are built from the two functions through \( z^{a} = \mathrm{exp}(a\,\mathrm{Ln}\,z) \) for \( z \ne 0 \), which for positive integer \( a = n \) reproduces the product of \( n \) copies of \( z \).[2](https://dlmf.nist.gov/4.2)
## Growth and decay models
The differential property gives the function its scientific role. A quantity whose rate of change is proportional to its current value satisfies \( y' = ky \), and its solutions are exactly the exponential functions \( y = y_{0}e^{kt} \).[3](https://openstax.org/books/calculus-volume-1/pages/6-8-exponential-growth-and-decay) With \( k > 0 \) this is exponential growth, applied to population growth and continuously compounded interest; with \( k > 0 \) in \( y = y_{0}e^{-kt} \) it is exponential decay, applied to populations that shrink, chemical compounds that break down, and radioactive decay.[3](https://openstax.org/books/calculus-volume-1/pages/6-8-exponential-growth-and-decay)
**Newton's law of cooling** fits the same framework after a change of variables. An object at temperature \( T \) in an ambient temperature \( T_{a} \) cools at a rate proportional to the difference, \( T' = -k(T - T_{a}) \); substituting \( y = T - T_{a} \) turns this into the standard decay equation \( y' = -ky \).[3](https://openstax.org/books/calculus-volume-1/pages/6-8-exponential-growth-and-decay)
A worked example shows the magnitudes involved. A bacterial population modeled by \( f(t) = 200e^{0.02t} \), with \( t \) in minutes, grows tenfold in 2 hours and reaches about 80,686 bacteria after 5 hours (300 minutes); solving \( 100{,}000 = 200e^{0.02t} \) gives \( t = \ln 500 / 0.02 \approx 310.73 \) minutes.[3](https://openstax.org/books/calculus-volume-1/pages/6-8-exponential-growth-and-decay) Because such models are continuous while real populations are discrete counts, model values must be interpreted in context.[3](https://openstax.org/books/calculus-volume-1/pages/6-8-exponential-growth-and-decay)
## History
The exponential function reached mathematics by two routes. The first was the logarithm, invented as a calculation aid: the Scottish baron [John Napier](https://www.edgepedia.org/john-napier) (1550–1617) and the Swiss craftsman Joost Bürgi (1552–1632) independently produced systems embodying the logarithmic relation and published tables for its use within years of one another.[6](https://old.maa.org/node/117597) Napier coined the term "logarithm" from Greek words meaning roughly "ratio-number", associating to each number another number whose sum properties reduced multiplication to addition.[7](https://www.open.edu/openlearn/science-maths-technology/mathematics-statistics/john-napier/content-section-5) Within about a century, what Napier had called a set of "excellent briefe rules" for calculation came to occupy a central place in theoretical mathematics.[7](https://www.open.edu/openlearn/science-maths-technology/mathematics-statistics/john-napier/content-section-5)
The second route was compound interest. In 1683, [Jacob Bernoulli](https://www.edgepedia.org/jacob-bernoulli) essentially found \( e \) while studying compound interest, evaluating the sequence \( (1 + 1/j)^{j} \) as \( j \to \infty \).[8](https://old.maa.org/node/117554) [Leonhard Euler](https://www.edgepedia.org/leonhard-euler) (1707–1783) later gave the constant its modern letter designation \( e \), discovered many of its properties, examined exponential and logarithmic functions as infinite series, and clarified the logarithms of negative numbers.[8](https://old.maa.org/node/117554)
## Computation
Software implementations of \( \mathrm{exp} \) do not sum the series directly over a wide range. The classic fdlibm algorithm used in Java's math library proceeds in three steps: first, argument reduction, writing \( x = k \cdot \ln 2 + r \) with \( |r| \le 0.5 \cdot \ln 2 \approx 0.34658 \); second, approximation of \( \mathrm{exp}(r) \) on that small interval by a degree-5 rational polynomial generated with a Remez algorithm, with maximum polynomial error bounded by \( 2^{-59} \); third, scaling back via \( \mathrm{exp}(x) = 2^{k} \cdot \mathrm{exp}(r) \).[4](https://code.googlesource.com/edge/openjdk/+/refs/heads/master/jdk/src/share/native/java/lang/fdlibm/src/e_exp.c) The documented error is under 1 ulp (unit in the last place), and among finite arguments only \( \mathrm{exp}(0) = 1 \) is exact.[4](https://code.googlesource.com/edge/openjdk/+/refs/heads/master/jdk/src/share/native/java/lang/fdlibm/src/e_exp.c)
Double-precision arithmetic bounds the usable range: for IEEE double, \( \mathrm{exp}(x) \) overflows when \( x > 709.78\ldots \) and underflows when \( x < -745.13\ldots \).[4](https://code.googlesource.com/edge/openjdk/+/refs/heads/master/jdk/src/share/native/java/lang/fdlibm/src/e_exp.c) Libraries also provide a companion function for small arguments: `expm1(x)` returns \( e^{x} - 1 \), and for values of \( x \) near 0 the exact sum of `expm1(x)` + 1 is much closer to the true value of \( e^{x} \) than computing `exp(x)` and subtracting 1, which loses digits to cancellation.[9](https://docs.oracle.com/en/java/javase/21/docs/api/java.base/java/lang/Math.html) MATLAB's `expm1` documentation illustrates the effect: for \( x = 10^{-15} \), `exp(X)-1` returns \( 1.1102 \times 10^{-15} \), a large relative error, while `expm1(X)` returns \( 1.0000 \times 10^{-15} \).[10](https://www.mathworks.com/help/matlab/ref/double.expm1.html)
## References
1. [Exponential Function, Wolfram MathWorld](https://mathworld.wolfram.com/ExponentialFunction.html)
2. [DLMF §4.2 Definitions: Logarithm, Exponential, Powers, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/4.2)
3. [6.8 Exponential Growth and Decay, Calculus Volume 1, OpenStax](https://openstax.org/books/calculus-volume-1/pages/6-8-exponential-growth-and-decay)
4. [fdlibm e_exp.c, OpenJDK source, Google Git](https://code.googlesource.com/edge/openjdk/+/refs/heads/master/jdk/src/share/native/java/lang/fdlibm/src/e_exp.c)
5. [Exponential function, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Exponential_function)
6. [Logarithms: The Early History of a Familiar Function, Clark & Montelle, MAA Convergence](https://old.maa.org/node/117597)
7. [John Napier: Napier's approach to logarithms, OpenLearn, The Open University](https://www.open.edu/openlearn/science-maths-technology/mathematics-statistics/john-napier/content-section-5)
8. [The Enigmatic Number e: A History in Verse and Its Uses in the Mathematics Classroom, Sarah Glaz, MAA](https://old.maa.org/node/117554)
9. [Math (Java SE 21 & JDK 21), Oracle documentation](https://docs.oracle.com/en/java/javase/21/docs/api/java.base/java/lang/Math.html)
10. [expm1: Compute exp(X)-1 accurately for small X, MATLAB Help Center](https://www.mathworks.com/help/matlab/ref/double.expm1.html)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Real analysis*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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