Logarithms and Exponential Models Lesson 4 2 Using

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 This lesson gives equation in the normal fashion us tools to be able § add to or subtract from both to sides manipulate the § multiply or divide both sides equations algebraically

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

Try It Out • Consider solution of 1. 7(2. 1) 3 x = 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

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

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? )

Assignment • Lesson 4. 2 • Page 164 • Exercises A § 1 – 41 odd

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

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. 071 t • By what percent does the population increase each year (What is the yearly growth rate)? § Use b = ek

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 C 14 present in his body remained. § The half life of C 14 is 5728 years § How long ago did the man die? • Use Q = A * ekt where A = 100% § Find the value for k, then solve for t

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

Did You Know?

Did You Know?

Did You Know?

Did You Know?

Assignment • Lesson 4. 2 • Page 164 • Exercises B § 43 – 57 odd