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Formulas and properties of logarithms.

Definition
The logarithm of number b on the base a (loga b) is defined as an exponent, in which it is necessary raise number a to gain number b (The logarithm exists only at positive numbers).

logab = x   if and only if   ax = b

The following types of logarithms are exist

  • loga b - logarithm of number b on the base a (a > 0, a ≠ 1, b > 0)
  • log b - common logarithms (Logarithms with base 10 are called common logarithms, a = 10).
  • ln b - natural logarithms (Logarithms with base e are called natural logarithms, a = e).

Formulas and properties of logarithms


For any a; a > 0; a ≠ 1 and any x; y > 0.
  1. alogab = b
  2. loga 1 = 0
  3. loga a = 1
  4. loga(x · y) = logax + logay
  5. loga xy = logax - logay
  6. loga 1x = -logax
  7. loga xp = p logax
  8. logak x = 1k loga x,    for k ≠ 0
  9. logax = logac xc
  10. loga x = logb xlogb a - change of base formula
  11. loga x = 1logx a

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