Maths / Logarithms
Logarithm Calculator (log, ln, log₂)
Calculate the logarithm of a number in base 10, e, 2, or any base, with a check and the change-of-base formula.
Logarithm Calculator (log, ln, log₂): The logarithm answers the question: to what power must the base be raised to give the number? log₁₀(1000) = 3 because 10³ = 1000. Any base works through the change-of-base formula, log_b(x) = ln(x) ÷ ln(b), and exact results such as log₂(1024) = 10 are shown as whole numbers. The check raises the base to the result to show it gives the number back. Runs 100% locally in your browser with zero server file uploads.
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log10(1,000) = 3
Check: 10 to the power 3 = 1,000
Change of base: log_b(x) = ln(x) ÷ ln(b).
Rules of logarithms
log(a × b) = log a + log b; log(a ÷ b) = log a − log b; log(aⁿ) = n × log a. These turn multiplication into addition, which is why logarithm tables and slide rules were used for calculation before calculators.
To measure sound levels in decibels, which are logarithmic, use the sound level meter; for other functions, the scientific calculator.
Logarithmic scales
Earthquake magnitudes, pH, and decibels are logarithmic: each step of 1 on the Richter or pH scale is a factor of 10, and every 10 dB is ten times the sound power.
How to use it
- Enter the number, greater than 0.
- Choose the base: 10, e for the natural log, 2, or another.
- Read the logarithm, with log₁₀, ln, and log₂ side by side.
Privacy & limitations
The calculation runs in your browser.
Related tools
Frequently asked questions
What is the difference between log and ln?
log usually means base 10, used for decibels and pH; ln is the natural logarithm, base e ≈ 2.71828, used for growth and decay. In computing, log often means base 2.
Why can't I take the log of 0 or a negative number?
No power of a positive base gives 0 or a negative number, so the logarithm is undefined there.
How do I undo a logarithm?
Raise the base to the result: if log₁₀(x) = 2.5, then x = 10^2.5 ≈ 316.2.
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