LESSON 24 (3.5) EXPONENTIAL AND LOGARITHMIC MODELS
(APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS)
You should learn to:
1. Use the Exponential Growth and Decay Model to solve real-life problems.
2. Use Exponential and Logarithmic Models to solve real-life problems.
3. Use the Logistic Growth Model to solve real-life problems.
Terms to know: exponential growth and decay model, logistic growth model, logistics curve , logarithmic model,
Exponential Growth & Decay: y  Cekt
C = initial amount
y = Amount at time t
t = time
k > 0 (growth)
k < 0 (decay)
Hint: Find the initial value, solve for k, write an equation in terms of y and t
Example 1: $500 is deposited into an account that pays 6.75% interest compounded continuously. How
long will it take the money to double in value?
a. Set up an exponential growth model equation, and find an exact solution for t .
1000  500e.0675t
1000 500 e.0675t

500
500
2  e.0675t
ln 2  ln e.0675t
ln 2  .0675t
ln 2
.0675 t

.0675 .0675
ln 2
t
.0675
b. Use a calculator to express the answer to 3 decimal place accuracy.
ln 2
.0675
t  10.269 years
t
Example 2: The world’s population for the years 1990 to 2007 can be approximated by the model
P  5400e.011852t , where t = 0 represents the year 1990 and P represents the population
in millions.
1990 is t  0
so, 2020 will be t  30
a. Use a calculator to estimate the world’s population in the year 2020. Show your set-up.
P  5400e.01185230
P  7705.690 million or
P  7, 705, 690, 000
b. According to the model, when will the world have a population of 10 billion people?
(Hint: 1 billion = 1000 million.) Show a graph which allows you to solve with your calculator.
y  5400e.011852 x
y  10, 000
find the point of intersection
x(or time) =52
So, 52 years from 1990,
or in 2042 the world population will be 10,000,000,000
Example 3:
In a research experiment, a population of 100 fruit flies is increasing according to the law of
exponential growth. After 2 days there are 300 flies. How many flies will there be after 5 days?
300  100e k 2
300 100 e k 2

100
100
3  e k 2
ln 3  ln e
ln 3  k  2
k 2
ln 3 k  2

2
2
k
ln 3
2
ln 3 5
y  100e 2
y  1558.846 so, about 1559 fruit flies
will be there in 5 days.
Example 4: The half-life of Carbon-14 is 5715 years.
a.
Find the value of k.
1
 e k 5715
2
ln 12  ln e
k 5715
ln 12  k  5715
ln 12
k  5715

5715
5715
k
ln(.5)
5715
b.
Write an exponential function modeling carbon-14 decay.
c.
Estimate the age of a fossil which contains one-third of its original Carbon 14.
1
3
ln(.5)
 e 5715
ln(.5)
y  Ce 5715
t
t
ln(.5)
ln 13  ln e 5715
t
ln 13 
ln(.5)
t
5715
(ln 13 ) 
5715 ln(.5)
5715

t 
ln(.5) 5715
ln(.5)
(ln 13 ) 
5715
t
ln(.5)
t  9058.061 so, about 9058 years old.
There are many examples of logarithmic models in the real-world. Sound intensity is measured in decibels (using a
logarithmic scale). pH (measure of acidity or alkalinity) is measured using a logarithmic scale. The Richter Scale for
measuring the intensity of earthquakes is a logarithmic scale. These are but a few common real-life logarithmic
models.
Example 6: On the Richter Scale, the magnitude R of an earthquake of intensity I is given by R  log I .
(I is the measure of the wave energy of the earthquake.)
a. Find the intensities of the San Francisco Bay area earthquake of 1989 which measured 7.1 on the
Richter Scale and the San Francisco earthquake of 1906 which measured 8.6 . Write each answer as a
power of 10.
1989 Earthquake
R  log I
1906 Earthquake
R  log I
7.1  log I
8.6  log I
107.1  I
108.6  I
b. Use your calculator to find out how many times as intense the earthquake of 1906 was.
108.6
 31.623 times as intense
107.1
Frequently, populations experience a period of rapid growth, followed by a declining rate of growth. Such populations
can often be represented by models known as Logistic Growth Models. The graph for such models is called a
logistics curve.
Example 7: On a college campus of 5000 students, one student returned from vacation with a contagious
virus. The spread of the virus through the student population is modeled by y 
5000
,
1  4999e 0.8t
where y is the total number of students infected after t days.
a. Enter the model on your calculator, build an appropriate
window, and graph the model. (Be careful to use parentheses
correctly (twice) in the denominator).
y  5000 / (1  4999e.8 x )
b. Use your model to determine how many students will be infected after 5 days.
2nd trace
value
x5
y  54.020 so, about 54 students infected
c. The college will cancel classes if 1000 or more of the students have contracted the virus. Does the college
need to cancel classes? If so, on what day?
y  1000
find the intersection
2nd trace
intersect
y  8.913 so, classes will be cancelled on the 9th day.
ASSIGNMENT 24
1.
$1000 is placed into a certificate of deposit (CD) in which interest is compounded continuously at a
rate of 3% per year. Use your calculator and the formula A  Pe rt to find:
a.
b.
c.
d.
2.
Show organized work on each of these problems. Then complete the textbook problems.
the amount that the CD would be worth in 1 year.
the amount that the CD would be worth in 5 years
the time it would take the CD to be worth $1,200.
the time it would take the CD to double in value.
Suppose 200 bacteria are introduced into a culture to study their rate of growth. Two days later, the
culture is found to contain 300 bacteria. Assume exponential growth  y  Cekt  .
a. Find the value of k.
b. Write the model (equation) for the population growth.
c. How many bacteria will be present in 3 more days (5 days after the start)?
3.
An isotope of carbon (C14 ) is used for estimating how long ago certain living organisms were on earth.
(The method is called carbon dating.) The half-life of C14 is approximately 5715 years. The skull of
an ancient primate originally contained 250 grams of C14 but now has only 25 grams remaining,
estimate how long ago the ancient primate lived (to the nearest thousand years).
4.
Find the half-life of a radioactive isotope if 4.92 grams out of an initial 5 grams of the isotope
remain after 10 years.
Pages 232-236 (1-6 (without using a calculator ), 15, 28, 39, 42 (use R  log I ), 49);
Pages 221-222: (35, 89) Page 212: (73, 89)