Logarithmic Functions

Properties of Logarithms

Properties of Logarithms

Logarithms have the following properties:

Since a ⁰ = 1 and a ¹ = a :

Property A: log_a1 = 0
Property B: log_a a = 1

Since a ^x and log_a x are inverses:

Property C: log_a a ^x = x
Property D: a ^log_ax = x

Since a ^p a ^q = a ^p+q and = a ^p-q :

Property E: log_a(pq) = log_a p + log_a q
Property F: log_a() = log_a p - log_a q

Since log_a(M ⁿ) = log_a(M·M·M ^... M) = log_a M + log_a M + log_a M + ^... + log_a M = n·log_a M

Property G: log_a(M ⁿ) = n·log_a 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: log_a M = , where b is any base.
Property H₁: log_a 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 log_a M by evaluating . For example, log₃4 = .

Example:

log₅10 + log₅20 - log₅8 = ?