1. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §9.4b Log Base-Change

2. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §9.4 → Logarithm Properties  Any QUESTIONS About HomeWork §9.4 → HW-46 9.4 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 3 Bruce Mayer, PE Chabot College Mathematics Summary of Log Rules  For any positive numbers M, N, and a with a ≠ 1

4. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.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-63_sec_9-4b_Log_Change_Base.ppt 5 Bruce Mayer, PE Chabot College Mathematics Change of Base Rule  Let a, b, and c be positive real numbers with a ≠ 1 and b ≠ 1. Then log b x can be converted to a different base as follows:

6. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 6 Bruce Mayer, PE Chabot College Mathematics Derive Change of Base Rule  Any number >1 can be used for b, but since most calculators have ln and log functions we usually change between base-e and base-10

7. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Evaluate Logs  Compute log 5 13 by changing to (a) common logarithms (b) natural logarithms  Soln

8. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 8 Bruce Mayer, PE Chabot College Mathematics  Use the change-of-base formula to calculate log 5 12. Round the answer to four decimal places  Solution Example  Evaluate Logs  Check 

9. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 9 Bruce Mayer, PE Chabot College Mathematics  Find log 3 7 using the change-of-base formula  Solution Example  Evaluate Logs Substituting into

10. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Swamp Fever

11. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Swamp Fever This does NOT = Log3

12. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 12 Bruce Mayer, PE Chabot College Mathematics Logs with Exponential Bases  For a, b >0, and k ≠ 0  Consider an example where k = −1

13. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  Evaluate Logs  Find the value of each expression withOUT using a calculator  Solution

14. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  Evaluate Logs  Solution:

15. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  Curve Fit  Find the exponential function of the form f(x) = ae bx that passes through the points (0, 2) and (3, 8)  Solution: Substitute (0, 2) into f(x) = ae bx  So a = 2 and f(x) = 2e bx. Now substitute (3, 8) in to the equation.

16. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 16 Bruce Mayer, PE Chabot College Mathematics Example  Curve Fit  Now find b by Taking the Natural Log of Both Sides of the Eqn  Thus the ae bx function that will fit the Curve

17. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 17 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §9.4 Exercise Set 70, 74, 76, 78, 80, 82  Log Tables from John Napier, Mirifici logarithmorum canonis descriptio, Edinburgh, 1614.

18. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 18 Bruce Mayer, PE Chabot College Mathematics All Done for Today Logarithm Properties

19. BMayer@ChabotCollege.edu MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 19 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –