1. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §9.4a Logarithm Rules

2. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §9.3 → Common & Natural Logs  Any QUESTIONS About HomeWork §9.3 → HW-45 9.3 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 3 Bruce Mayer, PE Chabot College Mathematics Product Rule for Logarithms  Let M, N, and a be positive real numbers with a ≠ 1, and let r be any real number. Then the PRODUCT Rule  That is, The logarithm of the product of two (or more) numbers is the sum of the logarithms of the numbers.

4. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 4 Bruce Mayer, PE Chabot College Mathematics Quotient Rule for Logarithms  Let M, N, and a be positive real numbers with a ≠ 1, and let r be any real number. Then the QUOTIENT Rule  That is, The logarithm of the quotient of two (or more) numbers is the difference of the logarithms of the numbers

5. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 5 Bruce Mayer, PE Chabot College Mathematics Power Rule for Logarithms  Let M, N, and a be positive real numbers with a ≠ 1, and let r be any real number. Then the POWER Rule  That is, The logarithm of a number to the power r is r times the logarithm of the number.

6. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 6 Bruce Mayer, PE Chabot College Mathematics Example  Product Rule  Express as an equivalent expression that is a single logarithm: log 3 (9∙27)  Solution log 3 (9·27) =log 3 9 + log 3 27. As a Check note that log 3 (9·27) = log 3 243 = 5 3 5 = 243 And that log 3 9 + log 3 27 = 2 + 3 = 5. 3 2 = 9 and 3 3 = 27

7. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Product Rule  Express as an equivalent expression that is a single logarithm: log a 6 + log a 7  Solution = log a (42). Using the product rule for logarithms log a 6 + log a 7 = log a (6·7)

8. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Quotient Rule  Express as an equivalent expression that is a single logarithm: log 3 (9/y)  Solution log 3 (9/y) =log 3 9 – log 3 y. Using the quotient rule for logarithms

9. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 9 Bruce Mayer, PE Chabot College Mathematics Example  Quotient Rule  Express as an equivalent expression that is a single logarithm: log a 6 − log a 7  Solution log a 6 – log a 7 = log a (6/7) Using the quotient rule for logarithms “in reverse”

10. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Power Rule  Use the power rule to write an equivalent expression that is a product: a) log a 6 − 3  Solution = log 4 x 1/2 Using the power rule for logarithms a) log a 6 − 3 = − 3log a 6 = ½ log 4 x Using the power rule for logarithms

11. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Use The Rules  Given that log 5 z = 3 and log 5 y = 2, evaluate each expression.  Solution

12. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Use The Rules  Solution  Soln

13. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  Use The Rules  Soln

14. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  Use The Rules  Express as an equivalent expression using individual logarithms of x, y, & z  Soln a) = log 4 x 3 – log 4 yz = 3log 4 x – log 4 yz = 3log 4 x – (log 4 y + log 4 z) = 3log 4 x –log 4 y – log 4 z

15. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  Use The Rules  Soln b)

16. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 16 Bruce Mayer, PE Chabot College Mathematics Caveat on Log Rules  Because the product and quotient rules replace one term with two, it is often best to use the rules within parentheses, as in the previous example

17. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 17 Bruce Mayer, PE Chabot College Mathematics Example  Expand by Log Rules  Write the expressions in expanded form  Solution a)

18. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Expand by Log Rules  Solution b)

19. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 19 Bruce Mayer, PE Chabot College Mathematics Example  Condense Logs  Write the expressions in condensed form

20. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 20 Bruce Mayer, PE Chabot College Mathematics Example  Condense Logs  Solution a)  Solution b)

21. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 21 Bruce Mayer, PE Chabot College Mathematics Example  Condense Logs  Solution c)

22. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Condense Logs  Solution d)

23. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 23 Bruce Mayer, PE Chabot College Mathematics Log of Base to Exponent  For any Base a  That is, the logarithm, base a, of a to an exponent is the exponent

24. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Log Base-to-Exp  Simplify: a) log 6 6 8 b) log 3 3 −3.4  Solution a) log 6 6 8 =8 8 is the exponent to which you raise 6 in order to get 6 8.  Solution b) log 3 3 − 3.4 = − 3.4

25. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 25 Bruce Mayer, PE Chabot College Mathematics Summary of Log Rules  For any positive numbers M, N, and a with a ≠ 1

26. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 26 Bruce Mayer, PE Chabot College Mathematics Typical Log-Confusion  Beware  Beware that Logs do NOT behave Algebraically. In General:

27. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 27 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §9.4 Exercise Set 24, 30, 36, 58, 60  Condense Logarithm

28. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 28 Bruce Mayer, PE Chabot College Mathematics All Done for Today Mathematical Association Log Poster

29. BMayer@ChabotCollege.edu MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 29 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –