Logarithmic Functions

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Properties of Logarithms

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Topics Properties of Logarithms

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Properties of Logarithms

Logarithms have the following properties:

Since a⁰ = 1 and a¹ = a:

Property A: log_a1 = 0
Property B: log_aa = 1

Since a^x and log_ax are inverses:

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

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

Property E: log_a(pq) = log_ap + log_aq
Property F: log_a() = log_ap - log_aq

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

Property G: log_a(Mⁿ) = n·log_aM

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_aM = , where b is any base.
Property H₁: log_aM =

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_aM by evaluating . For example, log₃4 = .