Core answer: log_b(x) asks "b to what power gives x": log₂(32) = 5 because 2⁵ = 32. Logs turn multiplication into addition: log(xy) = log x + log y — the trick that powered 300 years of slide rules and now measures earthquakes (Richter), sound (decibels), acidity (pH), and algorithm speed (O(log n)).

The three laws that run everything

LawFormulaExample
Productlog(xy) = log x + log ylog₂(8×4) = 3 + 2 = 5 = log₂32
Quotientlog(x/y) = log x − log ylog₁₀(1000/10) = 3 − 1 = 2
Powerlog(xᵏ) = k·log xlog₁₀(10²) = 2

Change of base: log_b(x) = ln x / ln b — your calculator's ln or log key computes any base. log₂(100) = ln100/ln2 ≈ 4.605/0.693 ≈ 6.64.

The special logs

  • log₁₀ (common): orders of magnitude; pH = −log[H⁺]; Richter magnitude.
  • ln (natural, base e ≈ 2.718): continuous growth/decay — radioactivity, cooling, compound interest in the limit.
  • log₂ (binary): computing — a sorted array of 1,000,000 items needs at most ⌈log₂1,000,000⌉ = 20 binary-search steps.

Worked examples

Example 1 — pH. Lemon juice at [H⁺] = 10⁻²·³ has pH = 2.3; each pH unit is a 10× acidity change, so coffee (pH 5) is ~500× less acidic.

Example 2 — Doubling time. ln 2 ≈ 0.693, so continuous growth at 7%/yr doubles in 0.693/0.07 ≈ 9.9 years — the Rule of 72's origin (72 = 100 ln 2, roughly).

Example 3 — Decibels. dB = 10·log₁₀(P/P₀): +10 dB = 10× power, +20 dB = 100×, +3 dB ≈ 2×. A 90 dB motorcycle is 1,000× the sound power of 60 dB conversation.

Common mistakes and myths

  1. log(x + y) ≠ log x + log y — the product law needs multiplication inside, not addition.
  2. log of ≤ 0 — undefined in reals: no real power of a positive base reaches 0 or negatives.
  3. Assuming base 10 everywhere — ln and log differ by factor ln 10 ≈ 2.303; mixing them wrecks calculations.
  4. Thinking logs shrink slowly always — log₂(10³⁰⁰) = 1,000, perfectly computable; that's why cryptography and big-number science work.
  5. Richter linearity myth — magnitude 7 vs 6 is not "10% stronger" but 10× the amplitude and ~31.6× the energy.