What this calculator does
This e calculator raises Euler's number, e, to any power you choose, and can also run the calculation backwards to find the natural log of a number. Euler's number is roughly 2.718281828, a fixed mathematical constant that turns up throughout calculus, compound growth and decay, and continuous probability, in the same way pi turns up in circles.
The most common confusion is expecting e to behave like a percentage or a growth rate you can adjust. It cannot: e is fixed, in the same way pi is fixed. What changes from problem to problem is the exponent x, which is why this calculator asks only for x (or for y, if you are solving the reverse problem of finding ln).
The formula
Enter an exponent x and the calculator raises e to that power (e to the power of x is written eˣ). Choose the reverse mode instead to find the natural logarithm of a number, which answers the question 'what power must e be raised to, to get this number back'. Because eˣ and ln(y) undo each other, each result includes a check figure running the other direction.
| Term | Meaning |
|---|---|
| e | Euler's number, approximately 2.718281828, a fixed mathematical constant. |
| eˣ | e raised to the power x, the base of natural (continuous) exponential growth. |
| ln(y) | The natural logarithm of y: the power e must be raised to in order to produce y. |
The inputs explained
| Field | What to enter |
|---|---|
| Calculate | Choose whether you want eˣ for a given exponent, or ln(y) for a given number. |
| Exponent x | The exponent to raise e to, used only in the eˣ mode. Can be negative or a decimal. |
| Number y (for ln) | The number to take the natural log of, used only in the ln mode. Must be greater than zero. |
When to use it
Continuous compound growth or decay
Continuous compounding, radioactive decay and some population models use eˣ directly rather than a repeated percentage step, because the growth is treated as happening continuously rather than in discrete periods.
Checking calculus and probability work
The exponential function eˣ and its inverse, the natural log, appear throughout calculus (as the function that is its own derivative) and in probability distributions such as the normal and exponential distributions.
Converting between a rate and a natural log figure
Some financial and scientific formulas quote a continuously compounded rate; converting that rate to or from a plain multiplier means computing eˣ or ln(y) directly, rather than approximating it.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What is e raised to a range of powers?
A range of exponents x, each raised as e to the power of x.
What is the natural log of a range of numbers?
A range of numbers y, each converted to its natural logarithm.
| Number y | ln(y) |
|---|---|
| 1 | 0 |
| 2.718281828 | 1.000000 |
| 7.389 | 2.0000 |
| 20 | 2.9957 |
| 100 | 4.6052 |
Questions
What exactly is e?
e is a fixed irrational constant, approximately 2.718281828, that arises naturally from continuous growth processes. It is the base of the natural logarithm and the unique number for which the function eˣ is its own derivative.
Is e the same thing as a percentage growth rate?
No. e is a fixed number, not a rate. A continuous growth rate of, say, 5% per year is applied as e^(0.05 × years), where the rate sits in the exponent rather than being e itself.
Why does e show up in compound interest?
As the number of compounding periods in a year increases towards continuous compounding, the compound-growth formula converges on eˣ. Ordinary compound interest calculators handle discrete periods (monthly, daily); e handles the continuous limit of that same idea.
What is the difference between ln and log base 10?
Natural log (ln) uses e as its base; common log (log₁₀) uses 10. They differ only by a constant multiplier (ln(y) = log₁₀(y) × ln(10)), but ln is the one that appears naturally in calculus and continuous-growth formulas.
For growth measured in discrete steps or percentages rather than continuous e-based growth, see the exponential growth calculator. For solving an exponential equation of the form a·bˣ = c, see the exponential equation calculator.