Logarithm Calculator
Calculate the logarithm of a number with any base. Enter the number and the base to find the answer instantly.
How to use the logarithm calculator
- Enter the number you want to take the logarithm of.
- Enter the base (use 2.71828 for the natural logarithm, ln).
- The calculator shows the logarithm automatically, plus a check calculation that confirms the answer.
How the logarithm is calculated
A logarithm answers the question: "what exponent must the base be raised to, to give this number?". The calculator uses the change-of-base formula to calculate logarithms with any base:
For example, log₁₀(100) = 2, because 10² = 100. The check field recalculates baseresult, which should give back the original number.
Why logarithms show up in so many scales
pH, decibels and the Richter scale have one thing in common: each step is a multiplication rather than an addition. An earthquake of magnitude 6 releases roughly thirty times more energy than one of magnitude 5, and pH 5 is ten times more acidic than pH 6. Such phenomena span so many orders of magnitude that a linear scale would be useless — the logarithm compresses them into numbers a person can reason about.
That also explains why the logarithm rules look the way they do. Because the logarithm turns multiplication into addition, log(a · b) equals log(a) + log(b), and log(aⁿ) becomes n · log(a). It was precisely this property that made logarithm tables and slide rules the standard tool for heavy calculation for several centuries: you could add instead of multiplying large numbers by hand.
Frequently asked questions
What is the natural logarithm (ln)?
The natural logarithm is the logarithm with base e (Euler's number, about 2.71828). It's widely used in mathematics, physics and economics, including continuous growth and decay. Enter 2.71828 as the base to calculate ln.
Why can't you take the logarithm of a negative number?
Because no real exponent can turn a positive base into a negative number (a positive number raised to any real exponent is always positive). The logarithm of zero or negative numbers is therefore undefined among the real numbers.
What are logarithms used for in practice?
Logarithms are used to measure sound level (decibels), earthquake strength (the Richter scale), pH in chemistry, and to analyze exponential growth and decay in economics and natural science.