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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Chabot Mathematics. §9.4b Log Base-Change. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. MTH 55. 9.4. Review §. Any QUESTIONS About §9.4 → Logarithm Properties Any QUESTIONS About HomeWork §9.4 → HW-46. Summary of Log Rules.

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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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  1. Chabot Mathematics §9.4bLog Base-Change Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. MTH 55 9.4 Review § • Any QUESTIONS About • §9.4 → Logarithm Properties • Any QUESTIONS About HomeWork • §9.4 → HW-46

  3. Summary of Log Rules • For any positive numbers M, N, and a with a≠ 1

  4. Typical Log-Confusion • Beware that Logs do NOT behave Algebraically. In General:

  5. Change of Base Rule • Let a, b, and c be positive real numbers with a ≠ 1 and b ≠ 1. Then logbx can be converted to a different base as follows:

  6. 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. Example  Evaluate Logs • Compute log513 by changing to (a) common logarithms (b) natural logarithms • Soln

  8. Example  Evaluate Logs • Use the change-of-base formula to calculate log512. • Round the answer to four decimal places • Solution  • Check

  9. Example  Evaluate Logs • Find log37 using the change-of-base formula • Solution Substituting into

  10. Example  Swamp Fever

  11. Example  Swamp Fever This does NOT = Log3

  12. Logs with Exponential Bases • For a, b >0, and k≠ 0 • Consider an example where k = −1

  13. Example  Evaluate Logs • Find the value of each expression withOUT using a calculator • Solution

  14. Example  Evaluate Logs • Solution:

  15. Example  Curve Fit • Find the exponential function of the form f(x) = aebx that passes through the points (0, 2) and (3, 8) • Solution: Substitute (0, 2) into f(x) = aebx • So a = 2 and f(x) = 2ebx . Now substitute (3, 8) in to the equation.

  16. Example  Curve Fit • Now find b by Taking the Natural Logof Both Sidesof the Eqn • Thus the aebx function that will fit the Curve

  17. WhiteBoard Work • Problems From §9.4 Exercise Set • 70, 74, 76, 78, 80, 82 • Log Tablesfrom John Napier, Mirifici logarithmorum canonis descriptio,Edinburgh, 1614.

  18. All Done for Today LogarithmProperties

  19. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

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