1. 8.5 – Using Properties of Logarithms

2. Product Property:

3. Product Property: log b mn

4. Product Property: log b mn = log b m

5. Product Property: log b mn = log b m + log b n

6. Ex.

7. Product Property: log b mn = log b m + log b n Ex. log 9 5x

8. Product Property: log b mn = log b m + log b n Ex. log 9 5x = log 9 5

9. Product Property: log b mn = log b m + log b n Ex. log 9 5x = log 9 5 + log 9 x

10. Product Property: log b mn = log b m + log b n Ex. log 9 5x = log 9 5 + log 9 x Quotient Property:

11. Product Property: log b mn = log b m + log b n Ex. log 9 5x = log 9 5 + log 9 x Quotient Property: log b m n

12. Product Property: log b mn = log b m + log b n Ex. log 9 5x = log 9 5 + log 9 x Quotient Property: log b m = log b m n

13. Product Property: log b mn = log b m + log b n Ex. log 9 5x = log 9 5 + log 9 x Quotient Property: log b m = log b m - log b n n

14. Product Property: log b mn = log b m + log b n Ex. log 9 5x = log 9 5 + log 9 x Quotient Property: log b m = log b m - log b n n Ex.

15. Product Property: log b mn = log b m + log b n Ex. log 9 5x = log 9 5 + log 9 x Quotient Property: log b m = log b m - log b n n Ex. log 9 x 9

16. Product Property: log b mn = log b m + log b n Ex. log 9 5x = log 9 5 + log 9 x Quotient Property: log b m = log b m - log b n n Ex. log 9 x = log 9 x 9

17. Product Property: log b mn = log b m + log b n Ex. log 9 5x = log 9 5 + log 9 x Quotient Property: log b m = log b m - log b n n Ex. log 9 x = log 9 x – log 9 9 9

18. Product Property: log b mn = log b m + log b n Ex. log 9 5x = log 9 5 + log 9 x Quotient Property: log b m = log b m - log b n n Ex. log 9 x = log 9 x – log 9 9 9 Power Property:

19. Product Property: log b mn = log b m + log b n Ex. log 9 5x = log 9 5 + log 9 x Quotient Property: log b m = log b m - log b n n Ex. log 9 x = log 9 x – log 9 9 9 Power Property: log b m p

20. Product Property: log b mn = log b m + log b n Ex. log 9 5x = log 9 5 + log 9 x Quotient Property: log b m = log b m - log b n n Ex. log 9 x = log 9 x – log 9 9 9 Power Property: log b m p = plog b m

21. Product Property: log b mn = log b m + log b n Ex. log 9 5x = log 9 5 + log 9 x Quotient Property: log b m = log b m - log b n n Ex. log 9 x = log 9 x – log 9 9 9 Power Property: log b m p = plog b m

22. Product Property: log b mn = log b m + log b n Ex. log 9 5x = log 9 5 + log 9 x Quotient Property: log b m = log b m - log b n n Ex. log 9 x = log 9 x – log 9 9 9 Power Property: log b m p = plog b m Ex.

23. Product Property: log b mn = log b m + log b n Ex. log 9 5x = log 9 5 + log 9 x Quotient Property: log b m = log b m - log b n n Ex. log 9 x = log 9 x – log 9 9 9 Power Property: log b m p = plog b m Ex. log 9 x 7

24. Product Property: log b mn = log b m + log b n Ex. log 9 5x = log 9 5 + log 9 x Quotient Property: log b m = log b m - log b n n Ex. log 9 x = log 9 x – log 9 9 9 Power Property: log b m p = plog b m Ex. log 9 x 7 = 7log 9 x

25. Ex. 2 Solve the following equations. a. 3 log 5 x – log 5 4 = log 5 16

26. Ex. 2 Solve the following equations. a. 3 log 5 x – log 5 4 = log 5 16 log 5 x 3 – log 5 4 = log 5 16

27. Ex. 2 Solve the following equations. a. 3 log 5 x – log 5 4 = log 5 16 log 5 x 3 – log 5 4 = log 5 16 log 5 x 3 = log 5 16 4

28. Ex. 2 Solve the following equations. a. 3 log 5 x – log 5 4 = log 5 16 log 5 x 3 – log 5 4 = log 5 16 log 5 x 3 = log 5 16 4

29. Ex. 2 Solve the following equations. a. 3 log 5 x – log 5 4 = log 5 16 log 5 x 3 – log 5 4 = log 5 16 log 5 x 3 = log 5 16 4 x 3 = 16 4

30. Ex. 2 Solve the following equations. a. 3 log 5 x – log 5 4 = log 5 16 log 5 x 3 – log 5 4 = log 5 16 log 5 x 3 = log 5 16 4 x 3 = 16 4 x 3 = 64

31. Ex. 2 Solve the following equations. a. 3 log 5 x – log 5 4 = log 5 16 log 5 x 3 – log 5 4 = log 5 16 log 5 x 3 = log 5 16 4 x 3 = 16 4 x 3 = 64 x = 4

32. b. log 4 x – log 4 (x – 6) = 2

33. log 4 x = 2 x – 6

34. b. log 4 x – log 4 (x – 6) = 2 log 4 x = 2 x – 6 Change to exponential form!!!

35. b. log 4 x – log 4 (x – 6) = 2 log 4 x = 2 x – 6 4 2 = x x – 6

36. b. log 4 x – log 4 (x – 6) = 2 log 4 x = 2 x – 6 4 2 = x x – 6 16 = x x – 6

37. b. log 4 x – log 4 (x – 6) = 2 log 4 x = 2 x – 6 4 2 = x x – 6 16 = x 1 x – 6

38. b. log 4 x – log 4 (x – 6) = 2 log 4 x = 2 x – 6 4 2 = x x – 6 16 = x x – 6 16(x – 6) = x

39. b. log 4 x – log 4 (x – 6) = 2 log 4 x = 2 x – 6 4 2 = x x – 6 16 = x x – 6 16(x – 6) = x 16x – 96 = x

40. b. log 4 x – log 4 (x – 6) = 2 log 4 x = 2 x – 6 4 2 = x x – 6 16 = x x – 6 16(x – 6) = x 16x – 96 = x 15x = 96

41. b. log 4 x – log 4 (x – 6) = 2 log 4 x = 2 x – 6 4 2 = x x – 6 16 = x x – 6 16(x – 6) = x 16x – 96 = x 15x = 96 x = 96/15

42. b. log 4 x – log 4 (x – 6) = 2 log 4 x = 2 x – 6 4 2 = x x – 6 16 = x x – 6 16(x – 6) = x 16x – 96 = x 15x = 96 x = 96/15 x = 32/5