Logarithm Calculator

Logarithms are the inverse of exponentiation. If bⁿ = x, then log_b(x) = n. They answer the question: “To what power must the base be raised to produce this number?”

Core definitions: logb(x) = n means bⁿ = x

Common logarithm types:

• Common log (log₁₀ or log): base 10, used in pH, decibels, Richter scale
• Natural log (ln): base e ≈ 2.71828, used in calculus, growth/decay, finance
• Binary log (log₂): base 2, used in computer science, information theory

Which bases are allowed? Any positive number except 1. Base 1 fails because every power of 1 is 1, so no exponent gets you anywhere else. Base 0 fails for the same reason in reverse. Negative bases fail because raising them to fractional powers leaves the real numbers entirely. And the input itself must be positive: a positive base raised to any real power is positive, so there is no real exponent that produces zero or a negative number.

Key logarithm rules:

Key values to memorize:

• log₁₀(10) = 1 | log₁₀(100) = 2 | log₁₀(1000) = 3
• ln(1) = 0 | ln(e) = 1 | ln(e²) = 2
• log₂(2) = 1 | log₂(4) = 2 | log₂(1024) = 10

Worked example, a decibel calculation: Sound intensity formula: dB = 10 × log₁₀(I ÷ I₀) Where I₀ = 10⁻¹² W/m² (threshold of hearing) Normal conversation (I = 10⁻⁶ W/m²): dB = 10 × log₁₀(10⁻⁶ ÷ 10⁻¹²) = 10 × log₁₀(10⁶) = 10 × 6 = 60 dB

Natural log in finance (continuous compounding): A = P × e^(r×t) → t = ln(A/P) ÷ r Time to double at 7% continuously: t = ln(2) ÷ 0.07 = 0.693 ÷ 0.07 = 9.9 years