Sometimes you know the base and the result of a power but not the exponent: how many times must 2 be doubled to pass 100? After how many years does 7% growth double your money? These questions are exponential equations of the form b^x = y, and logarithms solve them in one line. Enter the base and the target value, and the calculator finds x, checks it, and tells you when the answer is an exact fraction.
How to use the solve for exponent calculator
- Enter the base b. It must be positive and not 1; type
efor the natural base 2.71828…. - Enter the result y, which must be positive. Fractions like 1/8 and scientific notation like 1e9 are accepted.
- Read x on the tape. The steps show the logarithm method, and the items show which whole powers of b the answer falls between.
Formula
The derivation uses the power rule for logarithms, ln(bˣ) = x · ln b:
- Start with bˣ = y.
- Take the natural log of both sides: ln(bˣ) = ln y.
- Apply the power rule: x · ln b = ln y.
- Divide by ln b: x = ln y ÷ ln b.
Worked example
Solve 2ˣ = 100.
x = ln 100 ÷ ln 2 = 4.605170186 ÷ 0.6931471806 = 6.6438561898
Sanity check: 2⁶ = 64 and 2⁷ = 128, and 100 sits between them, so x must be between 6 and 7. ✓
Some exponents come out exact. For 8ˣ = 4, x = ln 4 ÷ ln 8 = 0.6666…, which the calculator recognizes as exactly 2/3: the cube root of 8 is 2, and 2² = 4. For 2ˣ = 1/8 the answer is exactly −3, since a negative exponent produces a reciprocal.
Real-world uses
Doubling time
Money growing at 7% a year is multiplied by 1.07 each year. Solving 1.07ˣ = 2 gives x = ln 2 ÷ ln 1.07 ≈ 10.24 years to double. The familiar “rule of 72” approximates the same answer as 72 ÷ 7 ≈ 10.3. The calculator’s “exponent per doubling” line gives this for any base above 1.
Half-life and decay
A quantity that keeps 0.5 of itself each period follows 0.5ˣ = y. To fall to 1/32 of the starting amount takes x = 5 half-lives. Because 0.5 is less than 1, larger exponents give smaller results, and a result above 1 (such as 0.5ˣ = 32) needs a negative exponent: x = −5.
Bits and digits
How many binary digits does it take to count to a million? Solve 2ˣ = 1,000,000: x ≈ 19.93, so 20 bits are needed. The same logic with base 10 tells you that a number y has ⌊log₁₀ y⌋ + 1 digits.
Tips
- Any log base works. log y ÷ log b, ln y ÷ ln b and log₂ y ÷ log₂ b all give the same x.
- Check the size first. Bracketing the answer between two whole powers catches typing errors quickly.
- Keep extra digits. Rounding ln b too early can shift the answer in the third or fourth decimal place, which matters for long time horizons.
For logarithms in general, including log₁₀, ln and log₂ side by side, use the logarithm calculator. To solve log_b(x) = y for the base or the argument instead, try the logarithm equation calculator.
Frequently asked questions
How do you solve for an exponent?
Take the logarithm of both sides of b^x = y. The power rule turns the left side into x·log b, so x = log y ÷ log b. Any logarithm base works as long as you use the same one on top and bottom.
Can the exponent be a fraction?
Yes. 8^x = 4 is solved by x = 2/3, because the cube root of 8 is 2 and 2² = 4. The calculator recognizes exact fractions like this and shows them alongside the decimal.
Why is there no solution when y is negative?
A positive base raised to any real power is always positive. No real x makes 2^x equal −8, so the equation has no real solution.
Why can't the base be 1, 0 or negative?
1 to any power is 1, so 1^x = y has either no solution or infinitely many. Zero and negative bases do not have real powers for most exponents, so logarithms with those bases are not defined.