Example 1
- Exponent: 1
≈ 2.7182818285
What is e¹? ≈ 2.7182818285.
Exponential Function Calculator
Enter the exponent x. The base is fixed: Euler’s number e ≈ 2.7182818285. The calculator computes eˣ = exp(x). e¹ ≈ 2.718. e⁰ = 1. e² ≈ 7.389. The result is always positive. This is not aˣ with a free base and not a growth model A₀·e^{rt} — only the value of exp(x).
Result
Enter an exponent x to see eˣ.
This page computes eˣ only, not 2ˣ or 10ˣ. The domain is ℝ, the range is (0, ∞). For a very large |x|, floating-point arithmetic may overflow — reduce |x|.
How it works
x1
expexp(1)
eˣ≈ 2.71828
The exponential function eˣ, also written exp(x), raises Euler’s number e to the power x. The base e ≈ 2.718281828459045… is a mathematical constant (NIST value). The calculator uses one field: the exponent x. The calculator computes exp(x). For x = 1 the result is e itself ≈ 2.7182818285. For x = 0 the result is 1. For x = 2 it is e×e ≈ 7.3890560989. For x = −1 it is 1/e ≈ 0.3678794412.
The domain is the set of real numbers ℝ. The range is the open interval (0, ∞): eˣ is never negative or zero. As x → −∞, eˣ → 0⁺. As x → +∞, eˣ → +∞. The inverse is the natural logarithm: ln(eˣ) = x and e^{ln y} = y for y > 0. Relation to an arbitrary base: for a > 0, aˣ = e^{x ln a}. Powers 2ˣ or 10ˣ belong on the power page — here the base is fixed.
The Maclaurin series gives the analytic construction: eˣ = Σn=0∞ xⁿ/n! = 1 + x + x²/2! + x³/3! + …. Differentiating term by term rebuilds the same series, so (eˣ)′ = eˣ. An equivalent limit is e = limn→∞ (1 + 1/n)ⁿ and eˣ = limn→∞ (1 + x/n)ⁿ. Those definitions are standard in the NIST Digital Library of Mathematical Functions, §4.2 (Exponential Function). Leonhard Euler popularized the symbol e in the 18th century; the constant itself already appears in the compound-interest limit (Bernoulli).
The result panel shows eˣ, an echo of x, the ln(eˣ) check, e⁻ˣ, the reciprocal 1/eˣ, a partial series sum (eight terms), and the constant e. The chart draws y = eˣ and marks the point (x, eˣ). A comma and a period in the x field are equivalent. For |x| roughly larger than about 700, exp leaves the floating-point range — the calculator reports overflow instead of Infinity.
Worked examples on the page: e¹ ≈ 2.71828, e⁰ = 1, e⁻¹ ≈ 0.36788, e² ≈ 7.38906, e^{1/2} ≈ 1.64872, e³ ≈ 20.08554, e⁻² ≈ 0.13534, and e^{ln 2} = 2 at x ≈ 0.693147. This calculator does not evaluate aˣ for arbitrary bases or a logarithm — only exp(x).
Formula
y = ex = exp(x), where x ∈ ℝ and y ∈ (0, ∞).
Series: ex = Σn=0∞ xn / n!. Limit: e = limn→∞ (1 + 1/n)n. Derivative: (ex)′ = ex. Inverse: ln(ex) = x. For a > 0: ax = ex ln a.
The symbol x is the exponent from the “Exponent x” field. The symbol e is the base ≈ 2.7182818285. Computation: exp(x).
Mathematical model: exponential function with base e (exp), following NIST DLMF §4.2.
Theory, series, and derivative
Definition
The exponential function exp : ℝ → (0, +∞) is given by exp(x) = eˣ, where e = 2.71828… is the base of the natural logarithm. Equivalent characterizations: the unique solution of y′ = y with y(0) = 1; the series Σ_{n=0}^∞ xⁿ/n!; the limit (1 + x/n)ⁿ as n → ∞. On this page the calculator evaluates exp(x), i.e. eˣ in floating-point arithmetic.
NIST DLMF §4.2–4.5 (Exponential Function) and Abramowitz & Stegun, Chapter 4, fix the notation eˣ = exp(x) and the identities with ln. MathWorld (Exponential Function) lists the same properties. The inverse is the natural logarithm: ln(exp(x)) = x and exp(ln y) = y for y > 0.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Historical and conceptual context
The number e arose in the seventeenth century with compound interest (Jacob Bernoulli) and in the calculus (Euler). The map eˣ is the prototype of exponential growth: populations, radioactive decay (e^{−λt}), continuous compounding. In complex analysis eᶻ = eˣ(cos y + i sin y) links the exponential to trigonometry (Euler’s formula); here we stay with real x.
Conceptually eˣ is not “exponentiation with an arbitrary base” aˣ — it is the special case a = e. For a > 0 one writes aˣ = exp(x ln a). Separate calculators on this site compute ln, log₁₀, and log_a.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Formal properties
Domain: ℝ. Codomain: (0, +∞). The function is strictly increasing and convex, with exp(0) = 1, exp(1) = e. Derivative and integral: (eˣ)′ = eˣ, ∫ eˣ dx = eˣ + C.
Identities: e^{x+y} = eˣ eʸ; e^{−x} = 1/eˣ; (eˣ)ʳ = e^{r x} for real r. Limits: eˣ → 0 as x → −∞; eˣ → +∞ as x → +∞. The Taylor series converges on all of ℝ.
Link to cosh/sinh: cosh x = (eˣ + e⁻ˣ)/2, sinh x = (eˣ − e⁻ˣ)/2.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Full derivation
Step 1 (series). Define E(x) = Σ xⁿ/n!. Termwise differentiation yields E′(x) = E(x), E(0) = 1.
Step 2 (uniqueness). If y′ = y and y(0) = 1, then (y e^{−x})′ = 0, so y = eˣ. Thus the series and the DE define the same function.
Step 3 (functional equation). Uniqueness for the DE implies e^{x+y} = eˣ eʸ.
Step 4 (inverse). Strict monotonicity gives a bijection ℝ → (0, ∞); the inverse is ln, with (ln y)′ = 1/y (NIST DLMF §4.2).
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Special cases
Values: e⁰ = 1; e¹ ≈ 2.71828; e² ≈ 7.38906; e⁻¹ ≈ 0.367879; e^{ln 10} = 10; e^{0.693147…} ≈ 2. The panel may show a partial series sum and the check ln(eˣ) ≈ x.
For x ≳ 709 floating-point arithmetic overflows; for x ≲ −745 the result shrinks to 0 (underflow). The calculator operates in the range of finite results.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Numerical and computational notes
exp is a library function (often with argument reduction and a minimax approximation). Comma and period are equivalent. Degrees do not apply — the argument is a real number, not an angle.
This field does not compute aˣ for a ≠ e — use exp(x ln a) or dedicated tools.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Related concepts and authorities
Related: ln, log₁₀, log_a, hyperbolic cosine, percentage change (exponential growth as a continuous model).
Authorities: NIST DLMF §4.2–4.5; Abramowitz & Stegun, Chapter 4; MathWorld — Exponential Function; Euler — Introductio. Educational tool.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
How to use
- Enter the exponent x in the Exponent x field, for example 1, 0, or −1.
- Click Calculate or wait for the live result. x = 1 gives ≈ 2.71828. x = 0 gives 1. x = 2 gives ≈ 7.38906.
- Check the panel: eˣ, the ln check, e⁻ˣ, the partial series sum, and the curve chart.
- Fractions and negatives are fine. e^{0.5} ≈ 1.64872. A very large |x| may leave the floating-point range.
- For another base (2 or 10) use the power page. Invert eˣ on the natural-logarithm page.
Symbol glossary
eˣ = exp(x) with base e. e¹ ≈ 2.71828. Domain ℝ, range (0, ∞).
- e
- Euler’s number ≈ 2.7182818285. Fixed base — not 2 and not 10. On the power page you enter the base yourself.
- x
- Exponent in the “Exponent x” field. 1, 0, 2, −1, or a fraction. A large negative x pulls the result toward zero from above.
- eˣ
- Primary result. Always positive. For x = 1 it is e; for x = 0 it is 1.
- n!
- Factorial in the denominator of the series term xⁿ/n!. Appears in the partial sum on the result panel.
Examples
Example 2
- Exponent: 0
1
What is e⁰? 1.
Example 3
- Exponent: −1
≈ 0.3678794412
What is e⁻¹? ≈ 0.3678794412.
Example 4
- Exponent: 2
≈ 7.3890560989
What is e²? ≈ 7.3890560989.
Example 5
- Exponent: 0.5
≈ 1.6487212707
What is e^{0.5}? ≈ 1.6487212707 (√e).
Example 6
- Exponent: 3
≈ 20.08553692
What is e³? ≈ 20.08553692.
Example 7
- Exponent: −2
≈ 0.1353352832
What is e⁻²? ≈ 0.1353352832.
Example 8
- Exponent: ≈ 0.693147
≈ 2
What is e^{ln 2}? ≈ 2.
Related calculators
Frequently asked questions
What is eˣ and how does it differ from aˣ?
eˣ = exp(x) is the exponential with base e ≈ 2.71828…. This calculator has one field: the exponent x. For an arbitrary base aˣ use the power page; the identity aˣ = e^{x ln a} holds for a > 0.
What are e¹, e⁰, and e²?
e¹ ≈ 2.7182818285 (e itself). e⁰ = 1. e² ≈ 7.3890560989. Those are examples 1, 2, and 4 on this page — the same results after “Fill this example”.
What are the domain and range of eˣ?
The domain is all real numbers ℝ. The range is (0, ∞): the result is always positive. As x → −∞ the value approaches 0 from above; as x → +∞ it grows without bound.
What are e⁻¹ and e to the 0.5?
e⁻¹ = 1/e ≈ 0.3678794412. e^{0.5} = √e ≈ 1.6487212707. Do not confuse √e with 0.5 × e.
Why is the derivative of eˣ equal to eˣ?
From the Maclaurin series eˣ = Σ xⁿ/n!, term-by-term differentiation shifts the index and rebuilds the same series. That is the defining property of exp; see NIST DLMF §4.2.
How does e relate to the limit (1+1/n)ⁿ?
A classical definition is e = lim (n→∞) (1 + 1/n)ⁿ. Equivalently eˣ = lim (1 + x/n)ⁿ. The approximation 2.71828… matches the value of e published by NIST.
Where do I invert eˣ?
On the natural logarithm: ln(eˣ) = x for every real x. Here the unknown is eˣ; there it is the exponent. ln e = 1, ln 1 = 0.
How is this different from 2ˣ or 10ˣ?
Here the base is fixed: e. 2¹⁰ = 1024 and 10² = 100 live on the power page, with a separate base field. This calculator always runs exp(x).
What if |x| is very large?
In floating-point arithmetic exp overflows when |x| is roughly larger than about 700. The calculator reports overflow or underflow instead of showing Infinity or 0.
Do the currency and unit settings in the page header affect the result?
No. The result depends only on the exponent x in the form field. The currency and unit settings in the page header do not affect this result.
What are e³ and e⁻²?
e³ ≈ 20.08553692. e⁻² ≈ 0.1353352832. Those are examples 6 and 7 on the page.
What does x ≈ 0.693147 give?
About 2, because ln 2 ≈ 0.693147. That is example 8: e^{ln 2} = 2 by the inverse definition.
How do I calculate e to the power of x?
Enter the exponent x. The result is exp(x) = eˣ with e ≈ 2.7182818285. For x = 1 you get ≈ 2.718; for x = 0 you get 1; for x = 2 ≈ 7.389.
What is e to the power of 2?
e² ≈ 7.3890560989. That is example 4 on this page. e³ ≈ 20.08553692; e⁻² ≈ 0.1353352832.
Can I change the base to 2 or 10?
Not on this page. The base is fixed at e. 2ˣ and 10ˣ are different functions — here the result is always eˣ = exp(x).
Is this a growth model A₀·e^{rt}?
No. The calculator returns only exp(x) for a single exponent. It does not multiply by an initial amount and does not solve exponential equations.
How do I invert eˣ?
The inverse is the natural logarithm: ln(eˣ) = x. At x ≈ 0.693147 the result eˣ ≈ 2, because e^{ln 2} = 2. Use a separate ln page when you need the inverse as the main output.
Sources
The definition, series, derivative, and constant e below come from recognized mathematical references.
- NIST DLMF §4.2 — Exponential Function
- NIST DLMF 4.2.1 — Series definition of exp
- MathWorld — Exponential Function
- Wikipedia — Exponential function
- Abramowitz, M.; Stegun, I. A. (eds.), Handbook of Mathematical Functions, NBS, 1964, Ch. 4 (exponential functions).
Page Updated November 2, 2026.