Calculate exp(x), find x from exp(x), and evaluate complex e^(a+bi) in radians or degrees, or graph y = e^x over any range of values.
Exp Calculator Formula
The calculator uses three formulas, one for each tab.
Real mode, forward direction:
exp(x) = e^x
Real mode, inverse direction:
x = ln(y)
Complex mode (Euler's formula):
exp(a + bi) = e^a * (cos(b) + i*sin(b))
- e: Euler's number, approximately 2.718281828459045.
- x: the real input exponent.
- y: the value of exp(x), always positive.
- a: the real part of a complex exponent.
- b: the imaginary part, in radians or degrees.
- ln: the natural logarithm, base e.
The Real tab evaluates ex directly or runs ln(y) to recover x when you already know exp(x). The Complex tab splits the exponent into a magnitude ea and a rotation angle b, then returns the rectangular form. The Graph tab samples exp(x) across the range you set and plots the curve so you can see how the output scales.
Reference Values and Range
Use these tables to sanity-check inputs before you trust a result.
| x | exp(x) | Notes |
|---|---|---|
| -5 | 0.006738 | Strong decay |
| -1 | 0.367879 | 1/e |
| 0 | 1 | Fixed point |
| 1 | 2.718282 | e |
| 2 | 7.389056 | e² |
| 5 | 148.4132 | Growth |
| 10 | 22026.47 | Five orders of magnitude |
| 709.78 | ~1.797e+308 | Upper limit before overflow |
| Complex input | exp result | Identity |
|---|---|---|
| 0 + πi | -1 | Euler's identity |
| 0 + (π/2)i | i | Quarter rotation |
| 0 + 2πi | 1 | Full rotation |
| 1 + 0i | e | Pure real |
Worked Examples
Example 1: Continuous interest. A balance grows at 4% per year compounded continuously for 7 years. The growth factor is exp(0.04 × 7) = exp(0.28). Enter 0.28 in the Real tab. The calculator returns about 1.32313. A starting balance of $5,000 becomes 5000 × 1.32313 = $6,615.65.
Example 2: Inverse problem. A bacteria culture grew by a factor of 12 under exponential growth. To find the exponent, switch the Real tab to "Find x from exp(x)" and enter 12. The result is x = ln(12) ≈ 2.4849. If the elapsed time was 3 hours, the rate constant is 2.4849 / 3 ≈ 0.828 per hour.
FAQ
What is the difference between exp(x) and e^x? Nothing. They are two notations for the same function. "exp" is the spelled-out form used in programming and science.
Why does the calculator cap x near 709.78? Above that value, ex exceeds the largest finite double-precision number (about 1.8 × 10308) and would return infinity.
Can exp(x) ever be negative or zero? No. For real x, exp(x) is always strictly positive. It approaches zero as x goes to negative infinity but never reaches it.
When should I use degrees instead of radians in the Complex tab? Use radians for math and physics work, since Euler's formula is defined that way. Use degrees only if your source data is already in degrees and you want the calculator to convert for you.
