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Mastering Logarithms for Solving Exponential Models

Learn to manipulate exponential equations algebraically by using logarithms. Solve problems involving growth rates, populations, and unsolvable exponential equations. Practice with examples and exercises.

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Mastering Logarithms for Solving Exponential Models

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  1. Logarithms and Exponential Models Lesson 4.2

  2. Using Logarithms • Recall our lack of ability to solve exponential equations algebraically • We cannot manipulate both sides of the equation in the normal fashion • add to or subtract from both sides • multiply or divide both sides This lesson gives us tools to be able to manipulate the equations algebraically

  3. Using the Log Function for Solutions • Consider solving • Previously used algebraic techniques(add to, multiply both sides) not helpful • Consider taking the log of both sides and using properties of logarithms

  4. Try It Out • Consider solution of1.7(2.1) 3x = 2(4.5)x • Steps • Take log of both sides • Change exponents inside log to coefficients outside • Isolate instances of the variable • Solve for variable

  5. Doubling Time • In 1992 the Internet linked 1.3 million host computers. In 2001 it linked 147 million. • Write a formula for N = A e k*t where k is the continuous growth rate • We seek the value of k • Use this formula to determine how long it takes for the number of computers linked to double 2*A = A*e k*t • We seek the value of t

  6. Converting Between Forms • Change to the form Q = A*Bt • We know B = ek • Change to the form Q = A*ek*t • We know k = ln B (Why?)

  7. Assignment • Lesson 4.2 • Page 164 • Exercises A • 1 – 41 odd

  8. Continuous Growth Rates • May be a better mathematical model for some situations • Bacteria growth • Decrease of medicine in the bloodstream • Population growth of a large group

  9. Example • A population grows from its initial level of 22,000 people and grows at a continuous growth rate of 7.1% per year. • What is the formula P(t), the population in year t? • P(t) = 22000*e.071t • By what percent does the population increase each year (What is the yearly growth rate)? • Use b = ek

  10. Example • In 1991 the remains of a man was found in melting snow in the Alps of Northern Italy. An examination of the tissue sample revealed that 46% of the C14 present in hisbody remained. • The half life of C14 is 5728 years • How long ago did the man die? • Use Q = A * ekt where A = 1 = 100% • Find the value for k, then solve for t

  11. Unsolved Exponential Problems • Suppose you want to know when two graphs meet • Unsolvable by using logarithms • Instead use graphing capability of calculator

  12. Did You Know?

  13. Did You Know?

  14. Did You Know?

  15. Did You Know?

  16. Assignment • Lesson 4.2 • Page 164 • Exercises B • 43 – 57 odd

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