1. T HE P OWER R ULE FOR L OGARITHMS

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

3. P ROOF OF THE P OWER R ULE FOR L OGARITHMS Let z = log b (x y ). Then b z = x y. Take the y’th root of both sides of the equation. b z/y = x Take the base b log of both sides. Log b (b z/y ) = log b (x) z/y = log b (x) z = y * log b (x) So log b (x y ) = y * log b (x)

4. E XAMPLE What is 2log(5) + 2log(2)?

5. S OLUTION By the power rule, 2log(5) = log(25) and 2log(2) = log(4). Thus, we have log(25) + log(4), which, by the product rule for logarithms, is log(100), which is 2.

6. E XAMPLE Given that log(2) =.69 and log(3) = 1.10, what is the value of log(36)?

7. S OLUTION Because 36 = 9 * 4, by the product rule of logarithms, log(36) = log(9) + log(4). By the power rule of logarithms, log(9) = 2log(3) and log(4) = 2log(2). Using the given values, log(36) = 2(0.69) + 2(1.1), which is equal to 3.58.