About this calculator
Enter a number and a base (10 by default, or type e for the natural logarithm) to find the logarithm. Switch to antilog to go the other way.
It also shows ln, log₁₀ and log₂ of your number together.
Worked examples
Real numbers, worked out by the same calculator. Press “Use these numbers” to try one above.
log₁₀ of 1,000
- log base 10 of 1,000
- 3
- Natural log (ln)
- 6.907755
- Common log (log₁₀)
- 3
- Binary log (log₂)
- 9.965784
The logarithm of 1,000 to base 10 is 3, meaning 10^3 = 1,000.
Show the working
- A logarithm asks: what power of the base gives the number? Here, 10 to the power of ? equals 1,000.
- Change of base: log_b(x) = ln(x) ÷ ln(b) = 6.907755 ÷ 2.302585 = 3.
log₂ of 64
- log base 2 of 64
- 6
- Natural log (ln)
- 4.158883
- Common log (log₁₀)
- 1.80618
- Binary log (log₂)
- 6
The logarithm of 64 to base 2 is 6, meaning 2^6 = 64.
Show the working
- A logarithm asks: what power of the base gives the number? Here, 2 to the power of ? equals 64.
- Change of base: log_b(x) = ln(x) ÷ ln(b) = 4.158883 ÷ 0.693147 = 6.
ln of 10 (base e)
- log base e of 10
- 2.302585
- Natural log (ln)
- 2.302585
- Common log (log₁₀)
- 1
- Binary log (log₂)
- 3.321928
The logarithm of 10 to base e is 2.302585, meaning e^2.302585 = 10.
Show the working
- A logarithm asks: what power of the base gives the number? Here, e to the power of ? equals 10.
- Change of base: log_b(x) = ln(x) ÷ ln(b) = 2.302585 ÷ 1 = 2.302585.
What a logarithm is
A logarithm answers the question “what power do I raise the base to, to get this number?” Since 10³ = 1,000, the log base 10 of 1,000 is 3. Logarithms turn multiplication into addition: log(ab) = log a + log b.
Changing the base
To find a logarithm in any base b from the natural logarithm, use log_b(x) = ln(x) ÷ ln(b). This calculator uses that rule, which is why it can handle any positive base other than 1.
Common bases
- Base 10 (common log): used for decibels, pH and the Richter scale.
- Base e ≈ 2.71828 (natural log, ln): used throughout calculus and science.
- Base 2 (binary log): used in computing and information theory.
Frequently asked questions
What is log of 100?
log₁₀(100) = 2, because 10² = 100.
Why can't I take the log of zero or a negative number?
No power of a positive base gives zero or a negative number, so the logarithm doesn't exist.
What is the natural logarithm?
The logarithm to base e (about 2.71828), written ln.
What is an antilogarithm?
The reverse of a logarithm: base raised to the power x. The antilog of 3 in base 10 is 1,000.
Formulas tested against hand-worked answers. Last reviewed 29 September 2026. These calculators do arithmetic only; they are not financial, tax or legal advice.