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Logarithm Calculator

Enter a number and a base. You get the logarithm to that base, and the natural, common and binary logarithms of the same number.

Your numbers

Result

How we calculate it

The logarithm of x to the base b is the power you must raise b to in order to get x. It is found with the change of base rule: log base b of x = ln(x) ÷ ln(b).

The natural log ln uses base e (about 2.71828), the common log uses base 10 and the binary log uses base 2. For 1000 the common log is 3, because 10³ = 1000. Its natural log is 6.9078 and its binary log is 9.9658. Results are shown to 10 significant digits.

Assumptions and limits

  • A logarithm needs a number above zero. Zero and negative numbers have no real logarithm.
  • The base must be above zero and not equal to 1.
  • Results very close to a whole number, such as 2.9999999999999996, are shown as that whole number.

Frequently asked questions

What is a logarithm?

It answers the question of how many times you multiply the base to reach a number. The log base 10 of 1000 is 3, because 10 × 10 × 10 is 1000.

What is the difference between log and ln?

Log usually means base 10, and ln means base e, the natural logarithm. This calculator shows both, plus base 2, and lets you choose any other base.

Where are logarithms used?

They are used for growth and decay, the pH scale, decibels, earthquake magnitude and how long money takes to double. The rule of 72 is an approximation based on the natural log of 2.

Related calculators

Worked example (a fixed example using the starting numbers, it does not change as you type)

Inputs: Number (above 0) = 1000; Base (above 0 and not 1) = 10.

  • Logarithm 3
  • Natural log (ln) 6.907755279
  • Common log (base 10) 3
  • Binary log (base 2) 9.965784285

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Last reviewed: . Formulas and assumptions are described on our methodology page. For information only; not professional advice.