Logarithm Calculator
Calculate the logarithm of a number in any base: common log (base 10), natural log (ln), log base 2 or a custom base.
Base
log₂(64)
6
Check: 2^6 = 64
- Natural logarithm (ln)
- 4.1588830834
- Common logarithm (log₁₀)
- 1.806179974
- Binary logarithm (log₂)
- 6
Frequently Asked Questions
The logarithm of x in base b is the power you must raise b to in order to get x. Because 2⁶ = 64, log₂(64) = 6. Logarithms turn multiplication into addition and are behind decibels, pH and the Richter scale.
log usually means the common logarithm, base 10 (log 1000 = 3). ln is the natural logarithm, base e ≈ 2.71828 (ln e = 1), used for continuous growth and in calculus. log₂ is the binary logarithm, base 2, used in computer science.
Use the change-of-base formula: log_b(x) = ln(x) / ln(b). For example, log₃(81) = ln 81 / ln 3 = 4. The calculator does this for any base greater than 0 other than 1.
A positive base raised to any power is always positive, so no power gives 0 or a negative number. The logarithm is therefore only defined for x > 0. As x gets closer to 0, its logarithm heads towards minus infinity.
log(a × b) = log a + log b, log(a ÷ b) = log a − log b, and log(aⁿ) = n × log a. In every base, log_b(1) = 0 and log_b(b) = 1.