How this calculator works
- Step 1Power10³ = 1000Raise the base to a power to get a number.
- Step 2Logarithmlog₁₀ 1000 = 3Ask which power of the base gives the number.
- Step 3Any baseln x ÷ ln bThe change-of-base formula handles every other base.
A logarithm answers one question: to what power must the base be raised to give this number? So log₁₀ 1000 is 3.
The calculator divides natural logs to handle any base, and shows ln, log₁₀ and log₂ of the same number alongside.
Worked example: log base 3 of 50
- Change of base
log_b(x) = log₁₀ x ÷ log₁₀ b
- xNumber 50
- bBase 3
- log₁₀ 50common log of 501.69897
- log₁₀ 3common log of 30.477121
- log₃ 501.69897 ÷ 0.4771213.560877
Check it: 3 to the power 3.560877 is 50. Any base works the same way, using either common or natural logs.
Positive, zero and negative logs
- Above 110log₂ 1024: 2¹⁰ = 1024.
- Exactly 10Any base to the power 0 is 1.
- Between 0 and 1−2log₅ 0.04: 5⁻² = 0.04.
Read the result
Logs count multiplications: log₁.₀₅ 2 = 14.206699, so about 14.2 steps of 5% double a value.
The answer is a power, not a number of the same kind as your input. Raising the base to that power gives your number back.
Assumptions and limits
- Number above 0Logs of 0 and negatives do not exist.
- Base not 1Every power of 1 is 1, so no answer fits.
- Exact powersBases 10 and 2 give whole answers exactly.
- DisplayRounded to six decimals.
Related powers and roots
A logarithm answers: to what power must the base be raised? Powers and roots are its partners.
To go the other way, use the exponent calculator. Logarithms also measure sound and acidity; see the decibel calculator and the pH calculator.
Common mistakes
Mistake: Writing log(a + b) = log a + log b.
Instead: The product rule is log(ab) = log a + log b.
Mistake: Mixing up ln and log.
Instead: ln is base e; log on most calculators is base 10.
Mistake: Taking the log of a negative number.
Instead: Logs need a number above 0.
Checking the method
Every golden case is re-worked bit by bit, by repeated squaring, without any library logarithm.
An independent script finds each logarithm in 60-digit arithmetic and compares it with the calculator to 14 significant digits.
Read how we test our calculators, or browse more maths calculators.