e Calculator - eˣ, Euler's Number Raised to Power x
Enter any value of x to compute eˣ, where e is Euler's number (approximately 2.71828). The calculator also solves the reverse problem: enter a result y and find ln(y), the natural logarithm. Switch between eˣ mode and ln(y) mode using the mode selector. Below the result you get a show-your-work Taylor series breakdown and a live curve chart of the exponential function across your chosen range.
Formula
Worked example
For x = 2: e^2 = 1 + 2 + 4/2 + 8/6 + 16/24 + ... = 1 + 2 + 2 + 1.333 + 0.667 + ... ≈ 7.38906. The reciprocal is e^(-2) = 1/7.38906 ≈ 0.13534. The square root is e^1 = e ≈ 2.71828.
What is e and why does it matter?
Euler's number e is the unique positive real number such that the function e^x is its own derivative. Its value is approximately 2.718281828459045, and like pi it is both irrational (cannot be expressed as a fraction) and transcendental (is not the root of any polynomial with rational coefficients). The number e was first studied systematically by Jacob Bernoulli in 1683 when he examined continuous compound interest, and it was Leonhard Euler who gave it the name and established its central role in analysis. The exponential function e^x is the only non-trivial function that equals its own derivative, which is why it appears naturally wherever growth or decay is proportional to the current amount: radioactive decay, population growth, compound interest, charging capacitors, Newton's law of cooling, and countless probability distributions.
How to compute eˣ: Taylor series and floating-point arithmetic
Modern computers compute e^x using a combination of argument reduction and polynomial approximation. The Taylor series e^x = 1 + x + x^2/2! + x^3/3! + ... converges for every real x. For small x (say |x| < 1) the series converges quickly: the error after n terms is smaller than |x|^n / n!. For large |x| the series converges slowly and direct summation accumulates rounding error, so hardware implementations use the identity e^x = e^(m + r) = e^m * e^r, where m is chosen so that e^m is tabulated exactly and |r| is small. This calculator uses IEEE-754 double precision (about 15-16 significant digits) via Math.exp(). The Taylor approximation shown in the steps panel is computed by this site purely for educational illustration.
eˣ versus ln(x): the inverse relationship
The natural logarithm ln(x) is the inverse of the exponential function: if e^a = b then ln(b) = a. This means e^(ln(x)) = x for every positive x, and ln(e^x) = x for every real x. The two functions are reflections of each other across the line y = x. Because of this symmetry, solving for an unknown exponent always involves the natural log: if you know the result y and want the exponent x, set x = ln(y). Switch to ln(y) mode in this calculator to do exactly that. Common applications include solving compound-interest equations, finding the half-life from a decay constant, and converting between exponential growth rates.
Properties and identities of eˣ
The exponential function satisfies several important algebraic identities. The addition law states e^(a+b) = e^a * e^b, so multiplying two exponentials is equivalent to adding their exponents. The subtraction law gives e^(a-b) = e^a / e^b. Raising an exponential to a power follows (e^a)^n = e^(a*n). For complex numbers x = a + bi the formula e^(a+bi) = e^a * (cos(b) + i*sin(b)) generalises the real exponential to the complex plane, yielding Euler's famous identity e^(i*pi) + 1 = 0 as the special case b = pi. The function is strictly positive for all real x: e^x > 0 always, and e^x approaches infinity as x increases and approaches zero from above as x decreases to negative infinity.
Common values of eˣ
| x | eˣ (exact form) | eˣ (decimal) | e^(-x) = 1/eˣ |
|---|---|---|---|
| -5 | e^(-5) | 0.006738 | 148.413 |
| -4 | e^(-4) | 0.018316 | 54.5982 |
| -3 | e^(-3) | 0.049787 | 20.0855 |
| -2 | e^(-2) | 0.135335 | 7.38906 |
| -1 | e^(-1) = 1/e | 0.367879 | 2.71828 |
| -0.5 | e^(-1/2) | 0.606531 | 1.64872 |
| 0 | e^0 = 1 | 1.000000 | 1.000000 |
| 0.5 | e^(1/2) = sqrt(e) | 1.64872 | 0.606531 |
| 1 | e^1 = e | 2.71828 | 0.367879 |
| 2 | e^2 | 7.38906 | 0.135335 |
| 3 | e^3 | 20.0855 | 0.049787 |
| 4 | e^4 | 54.5982 | 0.018316 |
| 5 | e^5 | 148.413 | 0.006738 |
| 10 | e^10 | 22026.5 | 0.0000454 |
| ln(2) = 0.693... | e^(ln 2) | 2.000000 | 0.500000 |
| ln(10) = 2.303... | e^(ln 10) | 10.00000 | 0.100000 |
Reference values of e raised to common exponents, rounded to six significant figures.
Frequently asked questions
What is eˣ?
e^x (read "e to the x" or "e raised to the power x") is the exponential function with base e, Euler's number (approximately 2.71828). It is the unique function that equals its own derivative, meaning the rate of change of e^x at any point equals its value at that point. It appears in compound interest, population growth, radioactive decay, probability theory, and almost every branch of science and engineering.
What is the value of e?
Euler's number e is an irrational mathematical constant approximately equal to 2.718281828459045. It cannot be expressed exactly as a fraction or a terminating decimal. It is defined as the limit of (1 + 1/n)^n as n approaches infinity, and it is the base of the natural logarithm. Its decimal expansion continues without repeating: 2.71828182845904523536...
What is the difference between eˣ and exp(x)?
They mean exactly the same thing. exp(x) is the standard computer-science and engineering notation for the exponential function e^x. Both denote e raised to the power x. The exp() notation is common in programming languages, spreadsheets (Excel, Google Sheets), and scientific literature to avoid ambiguity about the base.
What is the inverse of eˣ?
The inverse function of e^x is the natural logarithm, written ln(x) or log_e(x). If e^a = b, then ln(b) = a. This calculator provides an ln(y) mode: enter the result y and it returns the exponent x = ln(y). Note that ln is only defined for positive y, since e^x is always positive.
What is e^0?
e^0 = 1. Any positive number raised to the power zero equals 1, and Euler's number is no exception. This follows directly from the rule a^0 = 1 for any non-zero a, and from the Taylor series: e^0 = 1 + 0 + 0/2! + ... = 1.
What is e^1?
e^1 = e, Euler's number itself, approximately 2.718281828459045. Raising any number to the power 1 returns that number unchanged.
What is the Taylor series for eˣ?
The Taylor series expansion of e^x centered at zero is: e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ... = sum from k=0 to infinity of x^k / k!. The series converges for every real (and complex) number x, and converges faster when |x| is small. Adding more terms always improves the approximation. This calculator shows the partial sum with a user-chosen number of terms alongside the exact result so you can see how quickly the series homes in.
Why is eˣ its own derivative?
The exponential function e^x is the unique non-trivial function f such that f'(x) = f(x) for all x (with f(0) = 1). This property follows from the definition of e as the number for which the derivative of a^x equals a^x: only the base e achieves d/dx[a^x] = a^x exactly. In practical terms it means that the rate of growth of e^x always equals its current value - a hallmark of processes like unconstrained population growth or continuous compounding.