A logarithm answers the question: to what power must the base be raised to produce a given number? log_b(x) = y means b^y = x. For example, log₁₀(1000) = 3 because 10³ = 1000.

What is the difference between log and ln?

log (or log₁₀) uses base 10, commonly used in science and engineering. ln (natural log) uses base e ≈ 2.71828, commonly used in calculus, physics, and continuous growth models.

What is an antilogarithm?

The antilogarithm is the inverse of a logarithm. antilog_b(y) = b^y. For example, antilog₁₀(3) = 10³ = 1000.

Logarithm Calculator — log, ln, Custom Base

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log₁₀(1000) 3

Meaning 10³ = 1000

log₁₀(x) — Common Log 3

ln(x) — Natural Log 6.9078

log₂(x) — Binary Log 9.9658

log₁₀(x) — Custom Base 3

Antilog₁₀(x) — 10ˣ —

eˣ — Antilog (natural) —

What is a Logarithm?

A logarithm answers: to what power must the base be raised to get a given number? Formally, log_b(x) = y means b^y = x. Logarithms are the inverse of exponentiation.

Common Logarithm Types

1. Common Log (log₁₀)

Base 10. Used in science, decibels, pH scale, Richter scale.

2. Natural Log (ln)

Base e ≈ 2.71828. Used in calculus, continuous growth, physics.

3. Binary Log (log₂)

Base 2. Used in computer science, information theory, algorithms.

Key Properties

Product: log(a × b) = log(a) + log(b)
Quotient: log(a / b) = log(a) − log(b)
Power: log(aⁿ) = n × log(a)
Change of Base: log_b(x) = ln(x) / ln(b)
Identity: log_b(b) = 1 and log_b(1) = 0

Antilogarithm

The antilogarithm is the inverse: antilog_b(y) = b^y. For example, antilog₁₀(3) = 10³ = 1000.

Examples

log₁₀(1000) = 3 because 10³ = 1000
ln(e²) = 2 because e² = e²
log₂(256) = 8 because 2⁸ = 256
log₅(125) = 3 because 5³ = 125

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