Logarithm

The logarithm function is the inverse of the exponential function. The base "b" of the logarithm is positive (that is b>0) and different than 1. The most common bases are 10 and the mathematical constant e.PlanetMath.org: Logarithm WolframMathWorld: Logarithm
When the base is 10, log is used and when the base is e, ln (natural logarithm) or log is used. Notation: log_b(a) means logarithm in base "b" of "a".
We have log_b(x)=y if and only if x=b^y.

Example: log_2(8)=3 because 8=2^3.
log_2(1/32)=-5 because 1/32=2^(-5)

Note that the logarithm is defined only for positive numbers. That means whenever we write log_b(x), x should be positive.
We can say then that the domain of the logarithm is D=(0,∞)

A logarithm equation is an equation that involves logarithms, for example log_2(x+2)-log_2(x^2+4)=3

• These rules apply to logarithms with base b:
  • log(xy) = log(x) + log(y)
  • log(x/y) = log(x) - log(y)
  • log(x^y) = y * log(x)
  • (change of base) log_b(x)=log_a(x)/log_a(b)
  • log(1) = 0
  • log(b) = 1PlanetMath.org: Logarithm
  • Note that the logarithm is not additive, that means log(x+y) is not log(x)+log(y)

• JSTOR: Napier's Method as a Basis for the Theory of Logarithms

• Wolfram Demonstration Project: The Law of the Iterated Logarithm in Probability Theory