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Logarithmic Functions

Properties of Logarithms



Properties of Logarithms

Logarithms have the following properties:

Since a 0 = 1 and a 1 = a :

Property A: loga1 = 0
Property B: loga a = 1
Since a x and loga x are inverses:
Property C: loga a x = x
Property D: a logax = x
Since a p a q = a p+q and = a p-q :
Property E: loga(pq) = loga p + loga q
Property F: loga() = loga p - loga q
Since loga(M n) = loga(M·M·M ... M) = loga M + loga M + loga M + ... + loga M = n·loga M
Property G: loga(M n) = n·loga M

Property H

Logarithms have an additional property, called property H, and a property H1 that is a specific case of property H.

Property H: loga M = , where b is any base.
Property H1: loga M =

Applications of Properties

The numerous properties listed on this page can be used to evaluate logarithmic functions. Property H1 is especially useful when evaluating logarithms with a calculator: since most calculators only evaluate logarithms with base 10, we can evaluate loga M by evaluating . For example, log34 = .


log510 + log520 - log58 = ?

  = log5()  
  = log525  
  = log552  
  = 2.