📊 Math

Log Calculator — Free 2026

Compute logarithms in any base instantly — log base 10, natural log, log base 2, and custom bases.

Number Please enter a positive number.

Base

Results

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Log Base 10

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Natural Log (ln)

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Log Base 2

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Log Base 10

How It Works

Enter a number
Choose a base
Read the results

Understanding Logarithms

Logarithms are the inverse of exponentiation and answer a fundamental question: "to what power must a base be raised to get a given number?" If b^x = n, then log_b(n) = x. Logarithms transform multiplication into addition, which historically made complex calculations feasible before computers and remains vital in fields from acoustics (decibels) to seismology (Richter scale) to information theory (bits).

Common, Natural, and Binary Logarithms

The three most widely used logarithm bases each serve distinct purposes. Log base 10 (common log) is used in pH chemistry, decibel scales, and scientific notation. The natural logarithm (ln, base e ≈ 2.718) appears throughout calculus, differential equations, and continuous growth models. Log base 2 (binary log) is the language of computer science — it tells you how many bits represent a value and defines the complexity of algorithms like binary search.

Logarithm Properties and Rules

Key properties make logarithms powerful for simplifying expressions: log(a × b) = log(a) + log(b), log(a / b) = log(a) - log(b), and log(aⁿ) = n × log(a). The change of base formula, log_b(n) = log(n) / log(b), lets you convert between any bases — which is exactly what this calculator uses internally. For the inverse operation, try the exponent calculator, or explore roots with the square root calculator.

Frequently Asked Questions

What is a logarithm?

A logarithm is the inverse of exponentiation. It answers the question: to what power must a base be raised to produce a given number? For example, log base 10 of 100 is 2 because 10 raised to the power of 2 equals 100.

What is the difference between log and ln?

Log (without a specified base) typically means log base 10, also called the common logarithm. Ln is the natural logarithm, which uses the mathematical constant e (approximately 2.71828) as its base. Natural logarithms are fundamental in calculus, physics, and continuous growth models.

Why is log base 2 important?

Log base 2, also called the binary logarithm, is essential in computer science and information theory. It tells you how many bits are needed to represent a number. For example, log base 2 of 256 is 8, meaning you need 8 bits (one byte) to represent 256 different values.

Can you take the logarithm of a negative number?

In the real number system, logarithms are only defined for positive numbers. The logarithm of zero is undefined (negative infinity), and the logarithm of a negative number requires complex numbers. This calculator works with positive real numbers only.