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Logarithms and Exponential ModelsPowerPoint Presentation

Logarithms and Exponential Models

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## PowerPoint Slideshow about ' Logarithms and Exponential Models' - maxine-rivera

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### 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

This lesson gives us tools to be able to manipulate the 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 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

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

- Write a formula for N = A e k*t where k is the continuous growth rate

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?)

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.071t

- 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 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

Unsolved Exponential Problems

- Suppose you want to know when two graphs meet
- Unsolvable by using logarithms
- Instead use graphing capability of calculator

Assignment

- Lesson 4.2
- Page 164
- Exercises A
- 1 – 41 odd

- Exercises B
- 43 – 57 odd

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