1. The logarithm and its properties. The logarithm loga x is defined as the power to which the base a must be raised to equal x. 2. Several key properties of logarithms are defined, including: the logarithm in one base can be written in terms of another base; the logarithm of a number to the power of its exponent is that exponent times the logarithm of the number; and logarithms follow laws related to multiplication, division, and powers that parallel those operations with exponents.