Reverse an exponent: find the exponent with a logarithm or the base with a root from bˣ = y, or calculate the power or a constant growth rate.
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Reverse Exponent Formula
Choose exponent, base, power result, or the per-period growth rate from a starting value, ending value and number of periods.
Find the exponent when you know the base and the result:
Find the base when you know the result and the exponent:
Find the result when you know the base and exponent: y = b^x.
Find the growth rate when you know the start, end, and positive number of periods:
- b is the base.
- x is the exponent; it is the unknown in exponent mode.
- y is the result of the exponential expression.
- n is the number of periods in growth mode.
- r is the per-period growth rate as a decimal.
Exponent mode uses a positive base other than 1 and a positive result. Base mode returns the positive base for a positive result and notes the negative alternative for a nonzero even safe-integer exponent. A negative result is supported with a nonzero odd safe-integer exponent, including negative exponents. Growth mode requires positive initial and final values and positive periods.
In power-result mode, a zero base requires a positive exponent. A negative base is supported only with a safe integer exponent. Some exact rational powers of negative bases also have real values, but this calculator does not infer fractions such as 1/3 from rounded decimal entries or enumerate fractional negative-base roots.
Reference Tables
Use these to sanity check a result before you trust it.
| Equation | Solve for | Answer |
|---|---|---|
| 2^x = 32 | x | 5 |
| 10^x = 1000 | x | 3 |
| 5^x = 20 | x | 1.8614 |
| x^4 = 81 | base | 3 (also −3) |
| x^3 = 125 | base | 5 |
| x^5 = -32 | base | -2 |
| Total change | Periods | Per-period rate |
|---|---|---|
| Doubled (×2) | 5 | 14.87% |
| Doubled (×2) | 10 | 7.18% |
| Tripled (×3) | 10 | 11.61% |
| Halved (×0.5) | 5 | -12.94% |
| +50% | 5 | 8.45% |
Worked Example
Suppose you invested 1000 and ended with 1500 after 5 years. To find the annual rate:
- Divide: 1500 / 1000 = 1.5.
- Take the 5th root: 1.5^(1/5) = 1.0845.
- Subtract 1: 0.0845, or 8.45% per year.
Why does the exponent calculator reject base 1? For 1^x = 1, every real exponent works, so the answer is not unique. For 1^x equal to any other result, there is no solution.
Can the base be negative? For a negative target result, base mode supports a nonzero odd safe-integer exponent and returns the negative base. For a positive target and nonzero even safe-integer exponent, both positive and negative bases solve the equation; the positive base is primary and the negative alternative is shown. Power-result mode supports negative bases with safe integer exponents. Fractional negative-base powers are outside this calculator’s supported scope, even though some exact rational cases are real.
What if my result is between 0 and 1? That is fine. It just means the exponent is negative (when the base is greater than 1) or the rate represents decay rather than growth.
