5.4 Properties of Logarithms 3/1/2013

Properties of Logarithms Let m and n be positive numbers and b ≠ 1, Product Property Quotient Property Power Property

Expand the expression. Assume all variables are positive. Example 1 Expand a Logarithmic Expression Expand the expression. Assume all variables are positive. log7 y 3x b. a. log4 5x 2 SOLUTION a. log4 5x 2 = log4 5 log4 x 2 + Product property = log4 5 2 log4 x + Power property log7 y 3x b. = log7 3x log7 y – Quotient property log7 3 log7 x log7 y = + – Product property 3

Checkpoint Expand the expression. 5. log2 5x ANSWER log2 5 + log2 x 6. Expand and Condense Logarithmic Expressions Expand the expression. 5. log2 5x ANSWER log2 5 + log2 x 6. log 2x 3 ANSWER log 2 + 3 log x 7. log3 7 5x ANSWER log3 5 – + log3 x log3 7 8. log6 y 4x 2 ANSWER log6 4 – + 2 log6 x log6 y

5 𝑥 =125 (5 raised to what power equals 125?) 5 3 =125 x = 3 Example 2 Solve for x Find the value of x. a. log5 125 = x Rewrite in Exp. Form: 5 𝑥 =125 (5 raised to what power equals 125?) 5 3 =125 x = 3 Logx 64 = 3 b. Rewrite in Exp. Form: 𝑥 3 =64 (what do you raised to 3rd power to get 64?) 4 3 =64 x = 4 5

3 4 =𝑥 (3 to the 4th power equals what?) 81=𝑥 x = 81 Example 2 Solve for x Find the value of x. c. log3 x = 4 Rewrite in Exp. Form: 3 4 =𝑥 (3 to the 4th power equals what?) 81=𝑥 x = 81 ln x = 4 d. ln x = -1 e. Rewrite in Log. Form: log 𝑒 𝑥 =4 Rewrite in Exp. Form: 𝑒 4 =𝑥 x = 𝑒 4 Rewrite in Log. Form: log 𝑒 𝑥 =−1 Rewrite in Exp. Form: 𝑒 −1 =𝑥 x = 1 𝑒 6

Homework: WS 5.4 #1-7odd, 8-17 all,19-29 odd