A Logarithm Calculator

A logarithm asks what power a base must be raised to in order to produce a given number, and it turns multiplication into addition.

NumberMust be positive. Zero and negative numbers have no real logarithm.BasePositive and not equal to 1. Use 10 for common logs, 2 for doublings, or 2.718281828459045 for natural logs.

Logarithm in your base

Natural logarithm (base e): 6.9078
Common logarithm (base 10): 3
Check: base raised to the answer: 1,000
Equivalent doublings (base 2): 9.9658

Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.

The formula this calculator uses

log_b(x) = ln(x) / ln(b)

x: The number whose logarithm you want
b: The base, which must be positive and not equal to 1
ln: Natural logarithm, the logarithm to base e
log: Common logarithm, the logarithm to base 10

How to check the result by hand

1
Write the question as a power
Log base b of x asks what power of b gives x. For log base 10 of 1000 the question is ten to what power equals one thousand, and the answer is 3.
2
Check the argument is positive
Only positive numbers have real logarithms. If the number is zero or negative, stop and look for an error upstream, because no power of a positive base produces such a value.
3
Take the natural log of the number
Use the ln key. For 1000 the natural log is 6.907755278982137. This is an intermediate value and is not the answer unless the base happens to be e.
4
Take the natural log of the base
For a base of 10 the natural log is 2.302585092994046. The base's logarithm is always positive when the base is greater than 1, which is the usual case.
5
Divide and interpret
6.907755278982137 divided by 2.302585092994046 is 3. That is the logarithm. Sanity-check it by raising the base to that power and confirming you get back the original number.

For worked examples, common mistakes and the limits of this formula, read the full How To Calculate A Logarithm page.