1. Exponential and Logarithmic Models
Section 3.5
Exponential and Logarithmic Models

2. Objective By following instructions students will be able to:
Recognize the five most common types of models involving exponential or logarithmic functions.
Use exponential growth and decay functions to model and solve real-life problems.
Use Gaussian functions to model and solve real-life problems.
Use logistic growth functions to model and solve real-life problems.
Use logarithmic functions to model and solve real-life problems.
Fit exponential and logarithmic models to sets of data.

3. Exponential Growth Model

4. Exponential Decay Model

5. Gaussian Model

6. Logistic Model

7. Logarithmic Models

8. Example 1:
Estimates of the world population (in millions) from 1992 through 2000 are shown in the table. The scatter plot of the data is shown in the figure. An exponential growth model that approximates this data is Where P is the population (in millions) and t=2 represents Compare values given by the model with the estimates given by the U.S. Bureau of the Census. According to this model, what year corresponds to a world population of 6.5 billion?
Year
1992
1993
1994
1995
1996
1997
1998
1999
2000
Population
5445
5527
5607
5688
5767
5847
5926
6005
6083

9. Example 2:
In a research experiment, a population of fruit flies is increasing in accordance with the law of exponential growth. After 2 days there are 100 flies, and after 4 days there are 300 flies. How many flies will there be after 5 days?

10. Age of Dead Organic Material
Ratio of the content of radioactive carbon (carbon 14) to the content of nonradioactive carbon (carbon 12) is :

11. Example 3:
The ratio of carbon 14 to carbon 12 in a newly discovered fossil is Estimate the age of the fossil.

12. Example 4:
In 1997, the SAT math scores for college bound seniors roughly followed the normal distribution Where x is the SAT score for mathematics. Use a graphing utility to graph this function. From the graph, estimate the average SAT score.

13. Example 5:
On a college campus of 5000 students, one student returns from vacation with a contagious flu virus. The spread of the virus is modeled by
Where y is the total number infected after t days. The college will cancel classes when 40% or more of the students are ill.
How many students are infected after 5 days?
After how many days will the college cancel classes?

14. Example 6:
On the Richter scale, the magnitude R of an earthquake of intensity I is
Where Io=1 is the minimum intensity used for comparison. Find the intensities per unit of area for the following earthquakes. (Intensity is a measure of the wave of energy of the earthquake).
Tokyo and Yokohama, Japan in 1924; R=8.3
Izmit, Turkey, in 1999; R=7.4

15. Example 7:
The data in the table gives the yield y (in milligrams) of a chemical reaction after x minutes. Use a graphing utility to fit a logarithmic model to the data. X 1 2 3 4 5 6 7 8 Y
1.5
7.4
10.2
13.4
15.8
16.3
18.2
18.3

16. Example 8:
The total amount A (in billions of dollars) spent on health care in the United States in the years 1970 through 1996 are shown below. Find a model for the data and use the model to estimate the amount spent in In the list of data points (t,A), t represents the year, with t=0 corresponding to (Source: U.S. Health Care Financing Administrator) (0,74.3), (182.2), (2,92.3), (3,102.4), (4,115.9), (5,132.6), (6,151.9), (7, 172.6), (8,193.2), (9,218.3), (10, 247.3), (11, 291.4), (12, 328.2), (24, 360.8), (14, 396.0), (15, 428.7), (16, 466.0), (17, 506.2), (18, 562.3), (19, 623.9), (20, 699.5), (21, 766.8), (22, 836.6), (23, 895.1), (24, 945.7), (25, 991.4), (26, )

17. Revisit Objective Did we…
Recognize the five most common types of models involving exponential or logarithmic functions?
Use exponential growth and decay functions to model and solve real-life problems?
Use Gaussian functions to model and solve real-life problems?
Use logistic growth functions to model and solve real-life problems?
Use logarithmic functions to model and solve real-life problems?
Fit exponential and logarithmic models to sets of data?

18. Homework
Pg 266#s 1-41 ODD