Logarithm Calculator
Calculate logarithms in base 10, base e (natural log), base 2 or any custom base.
Enter your numbers to see the result.
What a logarithm answers
A logarithm asks: to what power must the base be raised to get this number? log₁₀(1000) = 3 because 10³ = 1000.
log_b(x) = y ⇔ b^y = x log_b(x) = ln(x) ÷ ln(b)
Useful rules
- log(a × b) = log a + log b
- log(a ÷ b) = log a − log b
- log(aⁿ) = n × log a
- log_b(1) = 0 and log_b(b) = 1
Where logs show up
- Finance: the time to reach a savings goal with compound interest is t = ln(goal ÷ start) ÷ ln(1 + r).
- Science: pH, decibels and the Richter scale are logarithmic; each step of 1 is a tenfold change.
- Computing: log₂ tells how many bits are needed or how many steps a binary search takes.
Frequently asked questions
What is ln?
The natural logarithm, a logarithm with base e ≈ 2.71828.
Can you take the log of a negative number?
Not with real numbers. Logarithms are defined only for positive inputs.
What is log 2 base 10?
About 0.30103.
Last reviewed October 2026. Results are estimates; check important decisions with a qualified professional.