Section 3.5
Modeling with Exponential
Logarithmic Functions
Exponential Growth and Decay
Example
The equation A=A 0 ekt models growth of a deer
population in a small local preserve. If the initial population
is 14 deer and the population grows to 20 in 4 years,
find the value of k, the rate of growth.
Logistic Growth Models
Example
If the following logistic equation models the number of people
who become infected with noro virus on a small cruise ship off
the coast of Alaska, how many people will become ill by the second
week if t is given in weeks.
f(t)=
500
1  e 1.5t
The Art of Modeling
Graphing Calculator - Exponential Regression
Press STAT then EDIT 1. Type in the values in List 1 (x values)
and List 2 (y values).
Then STAT, move over to CALC and press 0. You will get values
for a, b, r and r2. The values of a and b go into the equation
y=a*b^x or y=abx.
When you use a graphing utility the value of r MAY appear.
This r is called the correlation coefficient and is a measure of
how well the model fits the data. The value of r is such that
-1  r  1. A positive r means that as the x-values increase, so
do the y-values. A negative r means that as the x-values increase,
the y values decrease. The closer that r is to -1 or 1, the better the
model fits the data.
Example
L1
L2
1
8
2
18
3
49
4
123
5
312
6
780
Find the exponential regression equation on your calculator using
the data below. The data is for the population of black flies
which increases exponentially in a favorable lab environment.
L1 is the number of weeks and L2 is for the number of black
flies in the laboratory experiment.
See the next screen for continuation of example.
Example-Graphing Calculator
Continuing with this same data, we are now going to graph the equation
that we got earlier in this exercise and we are going to graph the data.
First make certain that your window is large enough so that you will
be able to see the graphed points.
Second, go to STAT PLOT and press ENTER on #1. Press ENTER
on the word "On", for type of graph use the first one, make Xlist: L1,
and Ylist:L2. Use either mark for the last option.
Now return to the graph and see the data.
Go to Y= and type in the equation that you got on the previous screen.
Return to the graph and observe how close the curve is to the actual data.
How can we obtain a logarithmic function that models
the data? A graphing utility can be used to obtain a logarithmic
model of the form y=a+b ln x. Because the domain of the
logarithmic function is the set of positive numbers, zero must
not be a value for x.
The technique for finding that logarithmic equation is the same
as for finding the exponential equation on a graphing calculator.
On the STAT, CALC menu choose 9 for Logarithmic Regression.
Expressing y=abx in Base e
Example
Rewrite y=3.82 (6.7) x in terms of base e.
A=A 0 e kt mod els growth of gray squirrels in a small city park.
If the initial population is 30 in 2000, and it grows to 38 in 1 year,
what is the growth rate, k?
(a)
.236
(b)
.023
(c)
.135
(d)
.126
1000
The equation f(x)=
models a rare type of flu
t
1  80e
which is infecting a town school. How many students, f(x),
will be sick in 2 weeks if t is time in weeks?
(a) 102
(b) 36
(c) 85
(d) 94