Logarithm Calculator
Calculate the logarithm of a number in base 10, base e, base 2 or any base you choose. Enter the number and pick the base; the common log, natural log and binary log are always shown side by side for comparison.
- log₁₀ (common)
- 3
- ln (natural)
- 6.90775528
- log₂ (binary)
- 9.96578428
Saved setups
Save a set of inputs you reuse — your usual rate, your loan, your room sizes — and load it back in one tap.
Your recent calculations
Results you calculate here are kept on this device so you can come back to them.
Formula
How to use it
- Enter a number greater than zero.
- Choose base 10, e or 2, or pick “Custom base” and type your own.
- Read the logarithm.
Worked examples
Common log of 1,000
- Logarithm
- 3
- ln (natural)
- 6.90775528
- log₂ (binary)
- 9.96578428
Natural log of 10
- Logarithm
- 2.30258509
Log base 5 of 125
- Logarithm
- 3
Rules of logarithms
Product: log(x × y) = log x + log y. Quotient: log(x ÷ y) = log x − log y. Power: log(xⁿ) = n × log x. In any base, log_b(1) = 0 and log_b(b) = 1.
Logarithms exist only for positive numbers, and the base must be positive and not equal to 1.
Where each base is used
Base 10 underlies the pH scale (pH = −log₁₀ of the hydrogen ion concentration) and decibels (10 × log₁₀ of a power ratio); on the Richter scale each whole number is a tenfold increase in measured amplitude.
Base e, written ln, appears in continuous growth and decay. Base 2 counts doublings and bits: log₂ 256 = 8, so 256 values fit in 8 bits.
Questions people ask
What is log base 10 of 1,000?
3, because 10³ = 1,000.
What is the natural log of 10?
About 2.302585.
What is log base 2 of 256?
8, because 2⁸ = 256.
What is the difference between log and ln?
“log” on a calculator means base 10 and “ln” means base e (about 2.71828). They differ by a constant factor: ln x = 2.302585 × log₁₀ x.