Math

Logarithm Calculator

Calculate logarithms in any base: common (log), natural (ln), binary (log2), or a custom base. See the formula and all standard log values.

Quick Answer

log_b(x) = y means b^y = x. Common log (base 10): log(100) = 2. Natural log (base e): ln(e) = 1. Binary log (base 2): log2(8) = 3. Use the change of base formula for any base: log_b(x) = ln(x) / ln(b).

Calculate Logarithm

About This Tool

The Logarithm Calculator computes logarithms in any base with high precision. Choose from common logarithm (base 10), natural logarithm (base e), binary logarithm (base 2), or enter any custom base. The tool shows the formula, result, and all standard logarithm values for quick reference.

Understanding Log Bases

The base of a logarithm determines the scale system. Base 10 maps well to our decimal number system and is used in scientific notation and decibel calculations. Base e (Euler's number, approximately 2.71828) appears naturally in calculus and continuous growth processes. Base 2 is essential in computer science where everything is binary.

Logarithm Properties

Key properties include: log(ab) = log(a) + log(b), log(a/b) = log(a) - log(b), log(a^n) = n*log(a), and log_b(1) = 0 for any base. These properties make logarithms powerful for simplifying multiplication into addition, which is why slide rules and logarithm tables were used before electronic calculators.

Frequently Asked Questions

What is a logarithm?
A logarithm answers the question: 'What exponent do I raise the base to in order to get this number?' log_b(x) = y means b^y = x. For example, log_10(1000) = 3 because 10^3 = 1000. Logarithms are the inverse of exponentiation.
What is the difference between log, ln, and log2?
log (common logarithm) uses base 10 and is used in science and engineering. ln (natural logarithm) uses base e (2.71828...) and is fundamental in calculus, physics, and continuous growth. log2 (binary logarithm) uses base 2 and is used in computer science and information theory. They are related by the change of base formula.
What is the change of base formula?
log_b(x) = ln(x) / ln(b) = log(x) / log(b). This lets you calculate any logarithm using any base. For example, log_5(25) = ln(25) / ln(5) = 3.2189 / 1.6094 = 2. Most calculators only have log and ln buttons, so this formula is essential for computing other bases.
Can you take the logarithm of a negative number?
Not with real numbers. Logarithms are only defined for positive real numbers. log(0) is undefined (negative infinity), and log of a negative number requires complex numbers. If you need log(-1), the result is i*pi in the complex number system, but this is beyond typical calculator use.
Where are logarithms used in real life?
Logarithms appear everywhere: the Richter scale (earthquake magnitude), decibels (sound intensity), pH scale (acidity), compound interest calculations, algorithms (binary search is O(log n)), music (octaves are logarithmic), and human perception (Weber-Fechner law says perception scales logarithmically with stimulus).

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