1. Properties of Logarithmic Functions Properties of Logarithmic Functions Objectives: Simplify and evaluate expressions involving logarithms Solve equations involving logarithms

2. Properties of Logarithms Product Property log b (mn) = log b m + log b n For m > 0, n > 0, b > 0, and b  1:

3. Example 1 = log 5 12 + log 5 10 log 5 120 = given: log 5 12  1.5440 log 5 10  1.4307 log 5 (12)(10)  1.5440 + 1.4307  2.9747

4. Properties of Logarithms Quotient Property For m > 0, n > 0, b > 0, and b  1: log b = log b m – log b n m n

5. Example 2 given: log 5 12  1.5440 log 5 10  1.4307 log 5 1.2 = log 5 12 – log 5 10  1.5440 – 1.4307  0.1133 = log 5 12 10

6. Properties of Logarithms Power Property For m > 0, n > 0, b > 0, and any real number p: log b m p = p log b m

7. Example 3 given: log 5 12  1.5440 log 5 10  1.4307 log 5 125 4 = 4  3 = 12 = 4 log 5 125 5 x = 125 5 3 = 125 x = 3

8. Practice Write each expression as a single logarithm. 1) log 2 14 – log 2 7 2) log 3 x + log 3 4 – log 3 2 3) 7 log 3 y – 4 log 3 x

9. Warm-Up Write each expression as a single logarithm. Then simplify, if possible. 4 minutes 1) log 6 6 + log 6 30 – log 6 5 2) log 6 5x + 3(log 6 x – log 6 y)

10. Properties of Logarithms Exponential-Logarithmic Inverse Property log b b x = x For b > 0 and b  1: and b log b x = x for x > 0

11. Example 1 Evaluate each expression. a) b)

12. Practice Evaluate each expression. 1) 7 log 7 11 – log 3 81 2) log 8 8 5 + 3 log 3 8

13. Properties of Logarithms One-to-One Property of Logarithms If log b x = log b y, then x = y For b > 0 and b  1:

14. Example 2 Solve log 2 (2x 2 + 8x – 11) = log 2 (2x + 9) for x. log 2 (2x 2 + 8x – 11) = log 2 (2x + 9) 2x 2 + 8x – 11 = 2x + 9 2x 2 + 6x – 20 = 0 2(x 2 + 3x – 10) = 0 2(x – 2)(x + 5) = 0 x = -5,2 Check:log 2 (2x 2 + 8x – 11) = log 2 (2x + 9) log 2 (–1) = log 2 (-1) undefined log 2 13 = log 2 13 true

15. Practice Solve for x. 1) log 5 (3x 2 – 1) = log 5 2x 2) log b (x 2 – 2) + 2 log b 6 = log b 6x

16. Solving Equations and Modeling Solving Equations and Modeling Objectives: Solve logarithmic and exponential equations by using algebra and graphs Model and solve real-world problems involving logarithmic and exponential relationships

17. Summary of Exponential-Logarithmic Definitions and Properties Definition of logarithmy = log b x only if b y = x Product Propertylog b mn = log b m + log b n Quotient Property Power Propertylog b m p = p log b m log b = log b m – log b n m n ( )

18. Summary of Exponential-Logarithmic Definitions and Properties Exp-Log Inverseb log b x = x for x > 0 log b b x = x for all x 1-to-1 for Exponentsb x = b y ; x = y 1-to-1 for Logarithms log b x = log b y; x = y Change-of-Base log c a = log b a log b c

19. Example 1 Solve for x. 3 x – 2 = 4 x + 1 log 3 x – 2 = log 4 x + 1 (x – 2) log 3 = (x + 1) log 4 x log 3 – 2 log 3 = x log 4 + log 4 x log 3 – x log 4 = log 4 + 2 log 3 x (log 3 – log 4) = log 4 + 2 log 3 x  –12.46 log 4 + 2 log 3 log 3 – log 4 = x

20. Example 2 Solve for x. log x + log (x + 3) = 1 log [x(x + 3)] = 1 10 1 = x(x + 3) x 2 + 3x – 10 = 0 (x + 5)(x – 2) = 0 x = 2,-5 10 1 = x 2 + 3x

21. Example 2 Solve for x. log x + log (x + 3) = 1 x = 2,-5 Let x = 2 log 2 + log (2 + 3) = 1 log 2 + log 5 = 1 1 = 1 log x + log (x + 3) = 1 Let x = -5 log -5 + log (-5 + 3) = 1 log -5 + log -2 = 1 undefined x = 2 Check:

22. Example 3 Solve for x. 8e 2x-5 = 56 e 2x-5 = 7 ln e 2x-5 = ln 7 2x - 5 = ln 7 x = ln 7 + 5 2 x  3.47

23. Example 4 Suppose that the magnitude, M, of an earthquake measures 7.5 on the Richter scale. Use the formula below to find the amount of energy, E, released by this earthquake.