Logarithm Calculator
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Calculate log base 10, natural log (ln), log base 2, or any custom base logarithm instantly.
Enter your values below and click Calculate for instant results.
A logarithm answers the question: what exponent do I need to raise this base to in order to get this number? The logarithm base 10 of 1000 is 3, because 10 to the power of 3 equals 1000. Logarithms are the inverse operation of exponentiation, just as subtraction is the inverse of addition.
Logarithms compress very large ranges of numbers into manageable scales. The Richter scale for earthquakes is logarithmic: an earthquake measuring 6 is ten times more powerful than one measuring 5, not just one unit stronger. Decibels for sound, pH for acidity, and the apparent brightness of stars all use logarithmic scales.
Log base 10, called the common logarithm, is the standard log used in most science and engineering calculations. The natural logarithm (ln) uses base e (approximately 2.71828) and appears extensively in calculus, physics, and continuous growth models. Log base 2 is used in computer science, information theory, and binary systems.
The logarithm of a product equals the sum of the logarithms of the factors. The logarithm of a quotient equals the difference of the logarithms. The logarithm of a power equals the exponent times the logarithm of the base. These properties allowed pre-calculator mathematicians to multiply and divide large numbers by adding and subtracting logarithms, which is the principle behind the slide rule.