Logarithm Calculator

A logarithm answers the question: “To what power must the base be raised to get a given number?”

logₐ(x) = y means aʸ = x

Common logarithm types:

• log₁₀(x) — common log (base 10)
• ln(x) — natural log (base e ≈ 2.71828)
• log₂(x) — binary log (base 2), used in computer science

Change of base formula:

logₐ(x) = ln(x) / ln(a) = log₁₀(x) / log₁₀(a)

Key properties:

• log(a × b) = log(a) + log(b)
• log(a / b) = log(a) - log(b)
• log(aⁿ) = n × log(a)
• log(1) = 0 for any base
• logₐ(a) = 1 for any base

Examples:

• log₁₀(100) = 2 because 10² = 100
• log₂(256) = 8 because 2⁸ = 256
• ln(e) = 1 because e¹ = e