e Calculator | eˣ | e Raised to Power of x

Enter an exponent x to compute eˣ (e raised to the power of x), using Euler's number e ≈ 2.718281828459045, plus the reciprocal e⁻ˣ and the equivalent percentage growth.

Quick Facts

Euler's number
e ≈ 2.718281828459045
An irrational, transcendental constant — the base of the natural logarithm.
Defining series
eˣ = 1 + x + x²/2! + x³/3! + …
The Taylor (Maclaurin) series that defines eˣ for every real x.
Self-derivative
d/dx [eˣ] = eˣ
eˣ is the only function (up to a constant multiple) equal to its own derivative.
Inverse relationship
ln(eˣ) = x
The natural logarithm undoes eˣ, and eˣ undoes ln(x) for x > 0.

Your Results

Calculated
eˣ
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e raised to the power of x
a · eˣ
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Scaled exponential value (coefficient × eˣ)
e⁻ˣ (1 / eˣ)
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Reciprocal — used for exponential decay
Growth over the period
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(eˣ − 1) × 100%, the % change for continuous growth/decay

Ready

Enter an exponent x, then press Calculate.

How the e Calculator | eˣ | e Raised to Power of x works

Euler's number e is a mathematical constant approximately equal to 2.718281828459045 — an irrational, transcendental number that is the base of the natural logarithm. Raising e to a power x gives the exponential function eˣ, one of the most important functions in mathematics because it is the unique function (up to a constant multiple) that equals its own derivative. This calculator evaluates eˣ, the scaled form a·eˣ, the reciprocal e⁻ˣ, and the equivalent percentage growth for a given exponent.

Formula and method

The exponential function is defined for every real number x by the infinite Taylor (Maclaurin) series eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + …, which always converges. In practice this calculator evaluates eˣ using the computer's built-in exponential routine (equivalent to Math.exp(x) in JavaScript), which returns the same value to full double-precision accuracy without you needing to sum the series by hand. The coefficient field extends the result to the general exponential model y = a·eˣ, the form used for continuous compound interest (A = Pe^(rt)), population growth, and radioactive decay (N(t) = N₀e^(kt)). The reciprocal e⁻ˣ = 1/eˣ is computed the same way with the sign of the exponent flipped, and the growth percentage is (eˣ − 1) × 100, the fractional change eˣ represents relative to a starting value of 1.

Common sources of error

  • Confusing eˣ with xᵉ: eˣ is an exponential function (fixed base e, variable exponent), while xᵉ is a power function (variable base, fixed exponent e ≈ 2.71828) — they are not the same and grow at very different rates.
  • Misreading "e" notation: the "e" in scientific/calculator notation (e.g., 2e5 = 2 × 10⁵) uses base 10, not Euler's number — don't mix the two meanings of the letter "e".
  • Assuming a negative exponent gives a negative result: eˣ is always positive for every real x; a negative x just makes the result a fraction between 0 and 1, never negative or zero.
  • Overflow for large x: eˣ grows extremely fast — for x beyond about 709.78 the true value exceeds the largest number a standard double-precision float can represent.

Checking your result

A few landmark values make eˣ easy to sanity-check: e⁰ = 1, e¹ ≈ 2.71828, e² ≈ 7.38906, and e⁻¹ ≈ 0.36788. Each time x increases by 1, eˣ multiplies by e (≈2.71828); each time x decreases by 1, it divides by e. You can also invert the calculation with the natural logarithm — ln(eˣ) should always return your original x — which is a fast way to confirm the tool computed the right value.

Applications

The exponential function eˣ underlies continuous compound interest (A = Pe^(rt), where r is the annual rate and t is time in years), exponential population or bacterial growth and radioactive/drug decay (N(t) = N₀e^(±kt)), and the bell-shaped normal distribution, whose probability density function contains e^(−x²/2). Because eˣ is its own derivative, it is also the natural building block for solving differential equations in physics, engineering, and finance wherever a quantity's rate of change is proportional to its current value.

Frequently Asked Questions

What is e in mathematics?
e (Euler's number) is a mathematical constant approximately equal to 2.718281828459045. It is the base of the natural logarithm and can be defined as the limit of (1 + 1/n)ⁿ as n approaches infinity, or as the sum of the infinite series 1 + 1/1! + 1/2! + 1/3! + …
How do you calculate eˣ?
eˣ is defined by the Taylor series eˣ = 1 + x + x²/2! + x³/3! + …, which converges for every real number x. This calculator evaluates it with a computer's built-in exponential function (equivalent to Math.exp(x) in JavaScript) for full double-precision accuracy.
What is the difference between eˣ and xᵉ?
eˣ is an exponential function with a constant base e and variable exponent x, while xᵉ is a power function with a variable base x and constant exponent e ≈ 2.71828. eˣ grows faster than any fixed power of x as x increases.
Where is eˣ used in real life?
eˣ appears in continuous compound interest (A = Pe^(rt)), exponential population growth and radioactive decay (N(t) = N₀e^(±kt)), and the normal distribution's probability density function. It is also the unique function equal to its own derivative, which is why it is central to differential equations.

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