1. Exponential and Logarithmic Data Modeling
Section 4.3
Exponential and Logarithmic Data Modeling

2. ANOTHER EXPONENTIAL GROWTH MODEL
Recall from Chapter 3, we used the natural (exponential) growth model of the form
P(t) = a · b t.
In the last section, we learned that eln b = b. So, we can write the natural growth model above as the continuous growth model
P(t) = a · b t = a · (eln b)t = a · e(ln b)t.
NOTE: ln b is the same as r in the formula
P(t) = a · e rt.

3. U. S. Census Population t (years) Year Act. Pop. (millions)
P(t) = 3.9 · e t E E2
1790
3.9
3.90
0.00
0.0000 10
1800
5.3
5.24
0.06
0.0036 20
1810
7.2
7.04
0.16
0.0256 30
1820
9.6
9.47
0.13
0.0169 40
1830
12.9
12.72
0.18
0.0324 50
1840
17.1
17.10

4. EXPONENTIAL MODEL
Definition: An exponential model is one of the form f (t) = a · e rt that describes either
An exponentially growing quantity if r is positive, or
An exponentially declining (decaying) quantity if r is negative.

5. EXPONENTIAL V. LOGARITHMIC MODELS
The exponential function ex is a rapidly increasing function of x, whereas
The logarithmic function ln x is a slowly increasing function of x.
Differences that are getting larger and larger suggest an exponential model.
Difference that are getting smaller and smaller suggest a logarithmic model.

6. EXAMPLES This is best modeled by an exponential model.
Change 1 22
1.5 23 2 25
2.5 29 4 3 35 6 x y
Change 5
16.0 15
19.0 3 25
20.5
1.5 35
21.5 1 45
22.3
0.8
This is best modeled by an exponential model.
This is best modeled by an logarithmic model.

7. EXAMPLE
The following table indicates the deaths from AIDS in the United States for men from 1988 to 1993 as reported by the U.S. Centers for Disease Control and Prevention
Year
1988
1989
1990
1991
1992
1993
AIDS deaths
(in thousands) 18 24 27 31 34 36
Find the best fit logarithmic model for this data. (Let x be number of years since 1987.) Use your model to predict the number of AIDS deaths in 1994.