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 of 1.7(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 his body 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?

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