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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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slide1

Chabot Mathematics

§9.4bLog Base-Change

Bruce Mayer, PE

Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

review

MTH 55

9.4

Review §
  • Any QUESTIONS About
    • §9.4 → Logarithm Properties
  • Any QUESTIONS About HomeWork
    • §9.4 → HW-46
summary of log rules
Summary of Log Rules
  • For any positive numbers M, N, and a with a≠ 1
typical log confusion
Typical Log-Confusion
  • Beware that Logs do NOT behave Algebraically. In General:
change of base rule
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:
derive change of base rule
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
example evaluate logs
Example  Evaluate Logs
  • Compute log513 by changing to (a) common logarithms (b) natural logarithms
  • Soln
example evaluate logs1
Example  Evaluate Logs
  • Use the change-of-base formula to calculate log512.
    • Round the answer to four decimal places
  • Solution

  • Check
example evaluate logs2
Example  Evaluate Logs
  • Find log37 using the change-of-base formula
  • Solution

Substituting into

example swamp fever1
Example  Swamp Fever

This does NOT = Log3

logs with exponential bases
Logs with Exponential Bases
  • For a, b >0, and k≠ 0
  • Consider an example where k = −1
example evaluate logs3
Example  Evaluate Logs
  • Find the value of each expression withOUT using a calculator
  • Solution
example curve fit
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.
example curve fit1
Example  Curve Fit
  • Now find b by Taking the Natural Logof Both Sidesof the Eqn
  • Thus the aebx function that will fit the Curve
whiteboard work
WhiteBoard Work
  • Problems From §9.4 Exercise Set
    • 70, 74, 76, 78, 80, 82
  • Log Tablesfrom John Napier, Mirifici logarithmorum canonis descriptio,Edinburgh, 1614.
all done for today
All Done for Today

LogarithmProperties

slide19

Chabot Mathematics

Appendix

Bruce Mayer, PE

Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu