Logarithms are evaluated by simplifying and rewriting until the expression is easily calculable or it's not possible to simplify the expression further. An example of a simplification is combining a sum or difference of logarithms into a single logarithmic expression.
Sums and differences of logarithms with the same base are combined into one expression using the following rules. First, the sum of two logarithms is equal to the logarithm of the product of the two arguments. Second, the difference of two logarithms is equal to the logarithm of the quotient of the two arguments.
Some logarithms can be removed from the expression. For instance, the logarithm of one with any base is always equal to zero. A logarithm that has the same base and argument is equal to one. Logarithms of scalars rather than variables can be rewritten as numerical values if the logarithm has a base of either 10 or e, which is approximately 2.718. To convert a logarithm to base 10 or e, it can be rewritten as the quotient of the logarithm base 10 or e of the argument and the logarithm of the original base. This quotient is equal to the logarithm base 10 or e of the original argument.