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Logarithms and Exponential Models

Logarithms and Exponential Models. Lesson 4.2. 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.

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Logarithms and 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. 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

  8. 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

  9. 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

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

  11. Assignment • Lesson 4.2 • Page 164 • Exercises A • 1 – 41 odd • Exercises B • 43 – 57 odd

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