4.1 Logarithms

Log definitions:

• log_b(x) is the number y, such that x = b^y. The b is called the base of the logarithm.
• lg(x) is log₂(x).

  Sometimes we use log(x) (neglecting to specify the base b) when we don't care what the base is. Logarithms satisfy this basic property:
  log_a(c) = log_a(b) log_b(c).

  
  Thus, when a and b are constants, log_a(x) and log_b(x) differ by the constant factor log_a(b). We needn't mention the base in contexts where we don't care about this factor.

  Also recall:

  • log_b(1) = 0,
  • log_b(b) = 1,
  • log(xy) = log(x) + log(y),
  • log(x^y) = y log(x),
  • log() is an increasing function: log(x) < log(y) iff x < y.
  • log(x) → ∞ (very slowly) as n → ∞.

  These approximations are very useful for back of the envelope estimates.
  10³ ≅ 2¹⁰, 10 = lg(kilo), 1/milli,
  10⁶ ≅ 2²⁰, 20 = lg(mega), 1/micro,
  10⁹ ≅ 2³⁰, 30 = lg(giga), 1/nano,
  10¹² ≅ 2⁴⁰, 40 = lg(tera), 1/pico,
  10¹⁵ ≅ 2⁵⁰, 50 = lg(peta),
  10¹⁸ ≅ 2⁶⁰, 60 = lg(exa).