1. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §9.5b Logarithmic Eqns

2. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §9.5 → Exponential Equations  Any QUESTIONS About HomeWork §9.5 → HW-47 9.5 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 3 Bruce Mayer, PE Chabot College Mathematics Summary of Log Rules  Solving Logarithmic Equations Often Requires the Use of Logarithms Laws  For any positive numbers M, N, and a with a ≠ 1

4. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 4 Bruce Mayer, PE Chabot College Mathematics Typical Log-Confusion  Beware  Beware that Logs do NOT behave Algebraically. In General:

5. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 5 Bruce Mayer, PE Chabot College Mathematics Exponent↔Logarithm Duality  Some Important Implications of the Properties of Logs & Exponents

6. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 6 Bruce Mayer, PE Chabot College Mathematics Solving Logarithmic Equations  Equations containing logarithmic expressions are called logarithmic equations.  We discussed previously how certain logarithmic equations can be solved by writing an equivalent exponential equation

7. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 7 Bruce Mayer, PE Chabot College Mathematics Solving Logarithmic Equations  Equations that contain terms of the form log a x are called logarithmic equations:  To solve a logarithmic equation rewrite it in the equivalent exponential form:

8. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Solve Logarithmic Eqn  Solve for x:  Solution:  Since the domain of logarithmic functions is positive numbers, the tentative solution must be checked

9. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 9 Bruce Mayer, PE Chabot College Mathematics Example  Solve Logarithmic Eqn  Check x = ½ ? ? ? ?  The Solution Set: {x| x = ½}

10. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Solve Logarithmic Eqn  Solve for x: log 2 (6x + 5) = 4  Solution: 6x + 5 = 2 4 6x = 11 log 2 (6x + 5) = 4 6x + 5 = 16 x = 11/6  The solution is x = 11/6. The check is left for us to do later

11. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 11 Bruce Mayer, PE Chabot College Mathematics Use Properties of Logarithms  Often the properties for logarithms are needed to solve Log Eqns.  The goal is to first write an equivalent equation in which the variable appears in just one logarithmic expression. We then isolate that expression and solve as in the previous example

12. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Solve Logarithmic Eqn  Solve for x: logx + log(x + 9) = 1  Solution x 2 + 9x = 10 log x + log (x + 9) = 1 log[x(x + 9)] = 1 x(x + 9) = 10 1 x 2 + 9x – 10 = 0 (x – 1)(x + 10) = 0 x – 1 = 0 or x + 10 = 0 x = 1 or x = –10

13. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  Solve Logarithmic Eqn  Solve for x: logx + log(x + 9) = 1  Check x = 1: 0 + log (10) log 1 + log (1 + 9) 0 + 1 = 1 TRUE  Check x = − 10: x = –10: log (–10) + log (–10 + 9) FALSE  The logarithm of a negative number is undefined. Thus the only solution is 1

14. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  Solve Logarithmic Eqn  Solve for x: log 3 (2x + 3) − log 3 (x − 1) = 2  Soln: log 3 (2x + 3) – log 3 (x – 1) = 2 log 3 [(2x + 3)/(x – 1)] = 2 (2x + 3)/(x – 1) = 3 2 (2x + 3)/(x – 1) = 9 (2x + 3) = 9(x – 1) x = 12/7 2x + 3 = 9x – 9  The Solution Set: {x| x = 12/7}

15. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  Solve Logarithmic Eqn  Solve:  Soln:

16. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 16 Bruce Mayer, PE Chabot College Mathematics Example  Solve Logarithmic Eqn  Soln cont.  Check x = 2: ? ? ? ? 

17. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 17 Bruce Mayer, PE Chabot College Mathematics Example  Solve Logarithmic Eqn  Check x = 3  The solution set is {x| x = 2, 3} ? ? ? ? 

18. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Solve Logarithmic Eqn  Solve:  Soln Product Rule Definition of Logarithms

19. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 19 Bruce Mayer, PE Chabot College Mathematics Example  Solve Logarithmic Eqn  Soln a. cont.  Check x = 2: ?  Logarithms are not defined for negative numbers, so x = 2 is not a solution. 

20. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 20 Bruce Mayer, PE Chabot College Mathematics Example  Solve Logarithmic Eqn  Check x = 5: ? ?   Thus The solution set is {x| x = 5}

21. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 21 Bruce Mayer, PE Chabot College Mathematics Example  Solve Logarithmic Eqn  Soln Quotient Rule Definition of Logarithms

22. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Solve Logarithmic Eqn  Check x = 4:   Thus The solution set is {x| x = 4} ? ? ?

23. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 23 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §9.5 Exercise Set 46, 56, 62, 66, 68, 74, 98  Exponent & Logarithm Laws.

24. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 24 Bruce Mayer, PE Chabot College Mathematics All Done for Today Solve Log-Eqn System

25. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 25 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

26. BMayer@ChabotCollege.edu MTH55_Lec-65_Fa08_sec_9-5b_Logarithmic_Eqns.ppt 26 Bruce Mayer, PE Chabot College Mathematics ReCall Logarithmic Laws  Solving Logarithmic Equations Often Requires the Use of the Properties of Logarithms