e^x Calculator
Compute the natural exponential. Enter any real exponent x — positive, negative, or decimal — and get e^x, its reciprocal e^(−x), and what the number means as continuous growth.
Example: with Exponent x 2 → e^x: 7.389056.
- Reciprocal e^(−x)0.135335
- Growth readingContinuous growth at rate 2 for one time unit multiplies the start by 7.389056 (+638.91%)
Computed by the calculator below using its default values. Change any input to see your own numbers.
e ≈ 2.718281828459045 (Euler's number). e^x is the unique function that equals its own rate of change — the reason it rules growth and decay models.
Why e, of all numbers
Take $1 growing at 100% a year. Compounded once you get $2; compounded quarterly, $2.44; monthly, $2.61; every instant, $2.71828… — that limit is e. The function e^x extends the idea: it is what continuous compounding at rate x for one unit of time does to a starting quantity. e² ≈ 7.389 means rate-2 growth multiplies you sevenfold.
What makes e^x special mathematically is that its slope equals its value everywhere: the function is its own derivative. Populations, radioactive decay, RC circuits, cooling coffee — anything whose rate of change is proportional to its current size runs on e^x, with negative exponents handling the shrinking cases (e^(−x) = 1/e^x).
How it’s calculated
y = e^x computed with Math.exp, where e = 2.718281828459045. Reciprocal: e^(−x) = 1/e^x. Growth reading: e^x − 1 expressed as a percent, the total change from continuous compounding at rate x for one time unit. Values beyond ±709 in the exponent overflow double precision (about 1.8 × 10^308) and are flagged.
Double-precision arithmetic carries about 15-16 significant digits, so extreme exponents are exact in magnitude but not in every trailing digit.
e^x landmarks
| x | e^x | Note |
|---|---|---|
| −1 | 0.3679 | 1/e |
| 0 | 1 | anything to the 0 |
| 0.6931 | 2 | x = ln 2, the doubling exponent |
| 1 | 2.7183 | e itself |
| 2 | 7.3891 | e squared |
| 5 | 148.4132 | — |
| 10 | 22,026.4658 | — |
Computed with y = e^x; ln 2 = 0.693147 is the exponent that exactly doubles.
Common mistakes
- Expecting e^(−x) to be negative — negative exponents make small positive reciprocals: e^(−2) ≈ 0.1353, never below zero.
- Confusing e^x with 10^x: e ≈ 2.718, so e^3 ≈ 20.1 while 10^3 = 1,000. Natural logs pair with e, common logs with 10.
- Reading the continuous rate as a simple percentage — continuous growth at rate 1 (100%) multiplies by e ≈ 2.718, not by 2.
- Entering the base instead of the exponent: this tool fixes the base at e and asks only for x.
Frequently asked questions
What is the e^x formula?
y = e^x, where e ≈ 2.718281828 is Euler's number. It is the continuous-compounding exponential: growth at rate x for one unit of time multiplies a quantity by e^x.
What is e squared?
e² ≈ 7.389056. Multiply e ≈ 2.718282 by itself. The calculator carries the full double-precision value and rounds the display to six decimals.
What does a negative exponent mean?
Decay. e^(−x) = 1/e^x, so e^(−0.5) ≈ 0.6065 means a quantity shrinking continuously at rate 0.5 keeps about 60.65% of its value after one time unit.
How is e^x different from 10^x or 2^x?
Only the base. Every exponential can be rewritten as e to something (10^x = e^(2.3026x)), but e is the natural choice in calculus because e^x is its own derivative — no correction factor appears.
What x makes e^x equal 2?
x = ln 2 ≈ 0.6931. That is why ln 2 shows up in doubling times and half-lives: continuous growth at rate r doubles in time (ln 2)/r.