Exponents and Logarithms

What are the properties of logarithms?

Logarithms have certain properties depending on interpretations of an equation. The following lists some of the most common properties (these rules are the same for all positive bases):

  • loga 1 = 0, because a0 = 1. For example, in the equation 140 = 1, the base is 14 and the exponent is 0. Because a logarithm is an exponent, this would mean the equation can be written as a logarithmic equation, or log14 1 = 0 (zero is the exponent).
  • loga a = 1, because a1 = a. For example, in the equation 31 = 3, the base is 3 and the exponent is 1; the result is 3, with the corresponding logarithmic equation being log3 3 = 1.
  • loga ax = x, because ax = ax. For example, 34 = 34, with the base as 3. The logarithmic equation becomes log3 34 = 4.


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