EXPONENTIAL MODELING AND LOGARITHMIC MODELING

Exponential Modeling and Logarithmic Modeling

Exponential Modeling and Logarithmic Modeling

Exponential Modeling

Question 1

During the 1980s the population of a certain city went from 100,000 to 205,000. Populations by Year are listed in the table below.

Year

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

Population in thousands

100

108

117

127

138

149

162

175

190

205

This data is approximated well by the exponential growth model P = 100 e0.08t, where t is the number of Years since 1980. In other words, the Year 1980 corresponds to t = 0, 1981 corresponds to t = 1, etc. The data points and model are graphed below.

Population data points and model P = 100 e0.08t

where t is number of years since 1980.

Problem 1: Use the model to predict the population of the city in 1994.

1991 corresponds to t = 11, so our model predicts that the population will be

P = 100 e0.08*11 = 241 thousand.

Problem 2: According to our model, when will the population reach 300 thousand?

To solve this problem we set 100 e0.08t equal to 300 and solve for t.

100 e0.08t = 300

e0.08t = 3 Take the natural logarithm of both sides.

ln e0.08t = ln 3

0.08t = ln 3

t = (ln 3)/0.08 = 13.73, approximately.

Therefore, the population is expected to reach 300 thousand about three fourths of the way through the year 1993.

It is important to recognize the limitations of this model. While it is obvious from the graph that for t between 0 and 9, the model values are very close to the actual population values, we should not assume that our model will give an accurate prediction of population for values of t much larger than 9. For instance, the model predicts that in the year 2080 (t = 100), the population of the city will be almost 300 million! That is not likely.

Suppose we know that a variable y can be expressed in the form aebx, but we don't know the values a and b. If we are given any two points on the graph of y, then it is possible to find the numbers a and b. The simplest case, and one that is often encountered in applications, is where we know the value of y when x = 0 and one other point on the graph of y.

Question 2

Assume that a population P is growing exponentially, so P = aebt, where t is measured in years. If P = 15000 in 1990, and P has grown to 17000 in 1993, find the formula for P.

Let t be the number of years since 1990. Then a = 15000, the value of P when t = 0. Note that t could have been chosen differently. For instance, we could let t be the number of years since 1900. But then we would not know the value of P when t = 0, so we would not know the value of a immediately.

We still need to find b. All we know about b is that it is positive, since the population is growing. Using the value we have found for a, we have

P = 15000 ...