4.5Modeling with
Exponential and
Logarithmic Functions
Discuss cell division.
Exponential Growth and Decay Models
The mathematical model for exponential growth or decay is given by
f (t) = A0ekt or A = A0ekt.
• If k > 0, the function models the amount or size of a growing entity. A0
is the original amount or size of the growing entity at time t = 0. A is the
amount at time t, and k is a constant representing the growth rate.
• If k < 0, the function models the amount or size of a decaying entity.
A0 is the original amount or size of the decaying entity at time t = 0. A is
the amount at time t, and k is a constant representing the decay rate.
y
y
increasing
A0
A0
y = A0ekt
k>0
decreasing
y = A0ekt
k<0
x
x
Ex: Given the exponential model to describe
the population of the indicated country, A,
in millions, t years after 2003:
India:
A  1049.7 e
Russia:
A  144.5 e
.015 t
 0.004 t
What is the initial population for each
country?
Which country is growing at a faster rate?
Ex cont: Given the exponential model to describe the population of
the indicated country, A, in millions, t years after 2003:
India:
Russia:
A  1049.7 e
A  144.5 e
.015 t
 0.004 t
What will Russia’s population be in 2006?
When will India’s population reach 1,600,000,000?
Example
• The value of houses in your neighborhood follows
a pattern of exponential growth. In the year 2006,
you purchased a house in this neighborhood. The
value of your house, in thousands of dollars, t
years after 2006 is given by the exponential
growth model V = 250e.07t
• How much did we pay for the house?
• Out of curiosity, how could we find the value in
2016?
• When will your house be worth $500,000?
(Which variable will we substitute? For which
will we solve?)
Cont…
Solution:
V = 250e.07t
In the year ____________our house should be worth $500,000.
p456 #26, 8
26) A logistic growth model for world population, f(x), in
billions, x years after 1949 is:
12.85
f ( x) 
1  4.21e
How many people populated the Earth initially (in 1949)?
How many people are expected in 2049?
What is the limiting size of the population that Earth will
eventually sustain (see # 26)? (If time 22 & 24)
 0.026 x
8. a) In 2000, the population of the Palestinians in the West
Bank, Gaza Strip, and East Jerusalem was approximately
3.2 million and by 2050 it is projected to grow to 12
million. Use the exponential growth model A  A0 e kt
in which t is the number of years after 2000, to find the
exponential growth function that models the data.
b) In which year will the Palestinian population be 9
million?
Text Example
•
•
Use the fact that after 5715 years a given amount of carbon14 will have decayed to half the original amount to find the
exponential decay model for carbon-14.
In 1947, earthenware jars containing what are known as the
Dead Sea Scrolls were found by an Arab Bedouin herdsman.
Analysis indicated that the scroll wrappings contained 76%
of their original carbon-14. Estimate the age of the Dead Sea
Scrolls.
Solution
We begin with the exponential decay model A = A0ekt. (ans. on next slide –
can you do it first?)
Text Example cont.
Solution
A0/2= A0ek5715
After 5715 years, A = A0/2
Divide both sides of the equation by A0.
Take the natural logarithm on both sides.
ln ex = x.
k = ln(1/2)/5715=-0.000121
Solve for k.
Substituting for k in the decay model, the model for carbon-14 is
A = A0e –0.000121t.
Now for part 2: In 1947, earthenware jars containing what are known as the Dead
Sea Scrolls were found by an Arab Bedouin herdsman. Analysis indicated
that the scroll wrappings contained 76% of their original carbon-14.
Solution
A = A0e-0.000121t
______ = A0e-0.000121t
Estimate the age of the Dead Sea Scrolls.
This is the decay model for carbon-14.
A = ___________ since 76% of the initial amount remains.
______ = e-0.000121t
Divide both sides of the equation by A0.
ln 0.76 = ln e-0.000121t
Take the natural logarithm on both sides.
ln ex = x.
t=ln(0.76)/(-0.000121)
Solve for t.
The Dead Sea Scrolls are approximately ___________ years old plus the
number of years between 1947 and the current year.
*Ex: Use the exponential growth model to
find the time it takes a population of 12
cells to quadruple in number at a growth
rate of 8% per second. Round your answer
to the nearest tenth.
Ex: (see notes on test review)
• Ex: Find how long it would take for a $2000
investment to double if it is invested at 14%
compounded continuously.
(see previous slide)
• Repeat the above to find how long it would
take that $2000 investment to double if it is
invested at 14% compounded bi-annually.
(Opt)Example
•
•
Find the exponential growth function
that models the data.
By what year will the population reach
40 million?
•
Use: A = A0
•
What is the original amount?
ekt
•
What is the initial time? The
final time given?
•
What is the final amount given?
Population (millions)
The graph below shows the growth of the Mexico City metropolitan area
from 1970 through 2000. In 1970, the population of Mexico City was 9.4
million. By 1990, it had grown to 20.2 million.
30
25
20
15
10
5
1970
1980
1990
2000
Year
Which variable in
our equation does
each of these
answers fill? What
information is
missing?
Example cont.
Solution
a. We use the exponential growth model
A = A0ekt
in which t is the number of years since 1970. This means that 1970
corresponds to t = 0. At that time there were 9.4 million inhabitants, so we
substitute 9.4 for _________ in the growth model:
We are given that there were 20.2 million inhabitants in 1990. Because
1990 is 20 years after 1970,
when ________ = 20 the value of _________ is 20.2. Substituting these
numbers into the growth model will enable us to find k, the growth rate. We
know that k > 0 because the problem involves growth.
We can now write:
Example cont.
Solution
20.2 = 9.4ek20
0.038 = k
We substitute 0.038 for k in the growth model to obtain the exponential
growth function for Mexico City. It is A = 9.4 e0.038t where t is measured in
years since 1970. What is our next step? (Remember our goal is to find
when the population will reach 40 million.)
Example cont.
Solution A
= 9.4 e0.038t
b. To find the year in which the population will grow to 40 million, we
substitute _______ in for _______ in the model from part (a) and solve for
________.
t = 38.10973065…
Because _________ is the number of years after 1970, the model indicates
that the population of Mexico City will reach 40 million by ____________.