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):

- log
_{a} 1 = 0, because *a*^{0} = 1. For example, in the equation 14^{0} = 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 log_{14} 1 = 0 (zero is the exponent).
- log
_{a} *a* = 1, because *a*^{1} = *a*. For example, in the equation 3^{1} = 3, the base is 3 and the exponent is 1; the result is 3, with the corresponding logarithmic equation being log_{3} 3 = 1.
- log
_{a} *a*^{x} = *x*, because *a*^{x} = *a*^{x}. For example, 3^{4} = 3^{4}, with the base as 3. The logarithmic equation becomes log_{3} 3^{4} = 4.