Logarithm Calculator (log, ln, log₂)

Logarithms Guide

What Logarithms Mean

log_b(x) = y means b^y = x. The logarithm answers the question 'what power do I raise b to, to get x?' log₁₀(1000) = 3 because 10³ = 1000. ln(e²) = 2 because e^2 = e². log₂(8) = 3 because 2³ = 8. Key values: log₁₀(10) = 1, log₁₀(100) = 2, log₁₀(0.1) = −1. ln(e) = 1, ln(1) = 0. log_b(1) = 0 for any base (b⁰ = 1). log_b(b) = 1 for any base (b¹ = b). Logarithms of numbers between 0 and 1 are negative.

Logarithm Laws

Product rule: log(ab) = log(a) + log(b). Quotient rule: log(a/b) = log(a) − log(b). Power rule: log(aⁿ) = n × log(a). Change of base: log_b(x) = log(x)/log(b) = ln(x)/ln(b). Example: log₃(81) = log(81)/log(3) = 1.908/0.477 = 4 (because 3⁴ = 81). These laws allow complex logarithmic expressions to be simplified and equations to be solved. The log laws are identical in structure to the index laws (log ab = log a + log b ↔ aⁿ × aᵐ = aⁿ⁺ᵐ).

Natural Log and e

The natural logarithm (ln) uses the irrational base e ≈ 2.71828. e arises naturally in continuous compound growth: if £1 is invested at 100% interest compounded continuously for 1 year, it grows to e = £2.718. e appears in: radioactive decay (N = N₀e^(−λt)), population growth, normal distribution probability density (e^(−x²/2)), Euler's formula e^(iπ) + 1 = 0. Derivatives: d/dx[eˣ] = eˣ (only function that is its own derivative — hence its ubiquity in differential equations). d/dx[ln x] = 1/x.

Logarithm Applications

pH = −log₁₀[H⁺] — each pH unit is a 10-fold change in hydrogen ion concentration. Decibels: dB = 10 × log₁₀(P/P₀) — the 10 dB rule means 10× the power, 20 dB means 100× the power. Richter scale: each unit is 10× the ground motion (30 in energy). Information theory: bits = log₂(number of possibilities) — a coin flip has log₂(2) = 1 bit of information. Computer science: O(log n) algorithms double their input to add one operation step. Music: pitch intervals are logarithmic — an octave always doubl