What a logarithm answers
A logarithm is the inverse of an exponent. It asks: to what power must the base be raised to produce this value? Since 10^3 = 1000, the log base 10 of 1000 is 3.
This calculator works in any base using the change-of-base formula, log_base(x) = ln(x) / ln(base). It divides the natural log of your value by the natural log of the base, which lets a single method handle base 2, base 10, base e, or anything else.
Reading the three results
The primary value is the log in the base you chose. Two standard logs are shown alongside it for convenience: the natural log (ln), which uses base e ≈ 2.71828, and the common log, which uses base 10.
A whole-number result means your value is an exact power of the base. A result between two integers means it falls between those powers. A log less than 0 means the value is smaller than 1 in that base.
Where logarithms appear
Logs turn multiplication into addition and compress huge ranges into manageable numbers.
- Measuring scales like decibels (sound), pH (acidity), and the Richter scale.
- Solving for an unknown exponent, such as how long an investment takes to double.
- Plotting data that spans many orders of magnitude on a readable axis.
- Estimating digit counts, since the number of digits relates to the base-10 log.
Common mistakes and limits
You cannot take the logarithm of zero or a negative number, because no real exponent produces a non-positive result. The value must be greater than zero.
The base must be positive and cannot equal 1, since 1 raised to any power is always 1 and could never reach another value. The calculator rejects these inputs rather than returning a misleading answer.
Formula
log_base(value) = ln(value) / ln(base)