Math & Statistics · 对数（Logarithm） · Updated 2026-10-25
What Is Logarithm?
Definition
log_b(a)=c ⟺ b^c=a: the log asks "what exponent on b gives a." Its core power is dimension reduction: log(ab)=log a+log b, log(a/b)=log a−log b, log(a^n)=n·log a — before calculators, engineers turned multiplication into addition with log tables. Common bases: ln (base e, native to calculus), log10 (orders of magnitude, decibels, pH), log2 (bits, algorithmic complexity). Richter magnitudes, decibels and pH are log scales — "+1" means 10×.
Formula
Change of base: log_b(a) = ln(a) / ln(b)
Worked Example
log2(1024)=10 since 2^10=1024; a pH-7 solution has H+ concentration 10^−7 mol/L and pH 5 is 100× more acidic (two orders); a magnitude-7 quake releases ~32× the energy of magnitude 6.
Common Misconception
Assuming log(a+b)=log a+log b — logs compress multiplication, not addition (correct: log(ab)=log a+log b). Also mixing log10 and ln: log10(100)=2 while ln(100)≈4.6 — same argument, different base, different value.
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