How this calculator works
log_b(x) answers “b to what power gives x?”. log₁₀(1,000) = 3 because 10³ = 1,000.
Any base is calculated with the change-of-base formula: log_b(x) = ln(x) ÷ ln(b).
The natural log, ln, uses base e ≈ 2.71828 and appears in growth, decay and interest formulas.
Logs are defined only for positive x, and the base must be positive and not 1.
Worked example
log base 10 of 1,000
- 10 to what power gives 1,000?
- Answer: 3.
- Check: 10³ = 1,000. Also ln(1,000) = 6.9078 and log₂(1,000) ≈ 9.966.
Questions people ask
Where are logarithms used?
pH in chemistry, decibels in sound, the Richter scale, compound interest and doubling times, and algorithm complexity in computing.
What are the log rules?
log(xy) = log x + log y, log(x/y) = log x − log y, and log(xⁿ) = n·log x.
How do I find how long money takes to double?
Years = ln(2) ÷ ln(1 + r). At 7%, ln 2 ÷ ln 1.07 ≈ 10.2 years.
Why is log of a negative number undefined?
No real power of a positive base produces a negative number or zero.
Last reviewed October 2, 2026