1. T HE P RODUCT R ULE FOR L OGARITHMS

2. W HAT IS THE P RODUCT R ULE FOR L OGARITHMS ? Simply stated, the product rule for logarithms is this: log b (xy) = log b (x) + log b (y) This rule can prove very useful for simplifying logarithms.

3. P ROOF OF THE P RODUCT R ULE Let M = b x and let N = b y. Then log b (M) = x and log b (N) = y. In addition, MN = b x+y by the properties of exponents. Taking the base b logarithm of both sides of our previous equation, we get log b (MN) = x + y. Substituting in log b (M) for x and log b (N) for y, we get log b (MN) = log b (M) + log b (N).

4. E XAMPLE We can use the product rule to solve many logarithm problems. For example, what is log 10 (125) + log 10 (8)? Using the product rule, log 10 (125) + log 10 (8) = log 10 (125 * 8) log 10 (125 * 8) = log 10 (1000) = 3.

5. E XAMPLE What is log 3 (6) – log 3 (2)? Using the product rule, log 3 (6) = log 3 (3) + log 3 (2). Thus, we have log 3 (3) + log 3 (2) – log 3 (2), which is equal to log 3 (3), which, of course, is 1.