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

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Chabot Mathematics. §9.4b Log Base-Change. Bruce Mayer, PE Licensed Electrical &amp; 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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Chabot Mathematics

§9.4bLog Base-Change

Bruce Mayer, PE

MTH 55

9.4

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

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

Substituting into

Example  Swamp Fever

This does NOT = Log3

Logs with Exponential Bases
• For a, b >0, and k≠ 0
• Consider an example where k = −1
Example  Evaluate Logs
• Find the value of each expression withOUT using a calculator
• Solution
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 Fit
• Now find b by Taking the Natural Logof Both Sidesof the Eqn
• Thus the aebx function that will fit the Curve
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

LogarithmProperties

Chabot Mathematics

Appendix

Bruce Mayer, PE