There are certain rules for combining logarithms. In the following cases, let a be a positive number that does not equal 0; n be a real number; and u and v be positive real numbers:

Logarithmic Rule 1: log_a(uv) = log_a(u) + log_a(v)

Logarithmic Rule 2: log_a(u/v) = log_a(u) - log_a(v)

Logarithmic Rule 3: log_a(u)ⁿ = nlog_a(u)

This can be expressed as follows: In rule one, multiplication inside the log is turned into addition outside the log (and vice versa); in rule two, division inside the log is turned into subtraction outside the log (and vice versa); and in rule three, an exponent on anything inside the log can be moved to the front of the log as a multiplier (and vice versa). But remember, these rules only apply if the bases are the same. For example, because the bases are not the same in log_a(u) + log_b(v), this expression can’t be simplified.