Logarithm Calculator
Free logarithm calculator. Calculate common logarithm (log base 10), natural logarithm (ln), binary logarithm (log base 2), and logarithms in any custom base with detailed explanations.
Compute the common logarithm (base 10) of a number.
What Is a Logarithm?
A logarithm is the exponent to which a base must be raised to produce a given number. It is the inverse operation of exponentiation. If bx = y, then logb(y) = x. Logarithms are used extensively in science, engineering, and computer science to simplify multiplication into addition and to measure quantities that span many orders of magnitude, such as sound intensity (decibels) and earthquake strength (Richter scale).
Types of Logarithms
Common logarithm (log): Base 10, widely used in science and engineering. Natural logarithm (ln): Base e (approximately 2.718), fundamental in calculus and growth models. Binary logarithm (log base 2): Base 2, essential in computer science and information theory. Custom-base logarithm: Any positive base other than 1, computed via the change-of-base formula.
Key Logarithm Properties
logb(xy) = logb(x) + logb(y) (product rule). logb(x/y) = logb(x) - logb(y) (quotient rule). logb(xn) = n * logb(x) (power rule). logb(b) = 1 and logb(1) = 0 for any valid base b.
Frequently Asked Questions
How do I calculate log base n?
Use the change-of-base formula: logn(x) = ln(x) / ln(n). This calculator handles it automatically when you select the custom-base tab.
Can I calculate logarithms of negative numbers?
In the real number system, logarithms are only defined for positive numbers. The logarithm of zero or a negative number is undefined.