MAT 171 Precalculus Algebra
Trigsted - Pilot Test
Dr. Claude Moore - Cape Fear Community College
CHAPTER 5:
Exponential and Logarithmic
Functions and Equations
5.1
5.2
5.3
5.4
5.5
5.6
Exponential Functions
The Natural Exponential Function
Logarithmic Functions
Properties of Logarithms
Exponential and Logarithmic Equations
Applications of Exponential and Logarithmic Functions
Objectives
·Solving Compound Interest Problems
·Exponential Growth and Decay
·Solving Logistic Growth Applications
·Using Newton’s Law of Cooling
Periodic Compound Interest Formula
A = Total amount after t years
P = Principal (original investment)
r = Interest rate per year
n = Number of times interest is compounded per year
t = number of years.
Find the Doubling Time: To find the time it will take for an investment to double
A becomes 2P.
How long will it taken (in years and months) for an investment to double if it earns 6.4%
compounded quarterly.
A = Total amount after t years = 2P
P = Principal =P
r = Interest rate per year = 6.4% = .064
n = Number of times interest is compounded per year = 4
t = number of years
Suppose an investment of $7500 compounded continuously grew to an amount of $8320
in 18 months. Find the interest rate and then determine how long it will take for the
investment to grow to $10000.
A = Total amount after t years = $8320
P = Principal= $7500
r = interest rate per year
t = number of years= 1.5
A = Total amount after t years = $10000
P = Principal= $7500
r = interest rate per year = .069
t = number of years
Exponential Growth
A model that describes the exponential uninhibited growth of a population, P,
after a certain time, t, is
P(t) = P0ekt
Wh
ere P0 = P(0) is the initial population and k > 0 is a constant called the relative
growth rate. ( Note: k is sometimes given as a percent.)
The population of a small town grows at a rate proportional to its current size. In 1925, the
population was 15000. In 1950, the population had grown to 25000. What was the population
of this town in 1970? Round to the nearest whole number.
Half Life
The required time for a given quantity of an element to decay to half of its
original mass.
Suppose that an item is found containing 7% of its original amount of carbon.
If the half life of that element is 25 years, how old is the item?
A0 is the original amount of carbon.
A(25)= ½ A0 because the half life of the item is 25 years.
Continued on the next page
Suppose that an item if found containing 7% of its original amount of carbon. If
the half life of that element is 25 years, how old is the item?
The item is about 96 years old.
Logistic Model
Describes population growth when outside limiting factors that
affect population growth exist.
y=C
Logistic Growth
A model that describes the logistic growth of a population P at any
time t is given by the function
Where B,C and k are constants with C > 0 and k < 0.
Six fish were introduced into a small backyard pond. Because of limited food, space, and
oxygen, the carrying capacity of the pond is 100 fish. The fish population at any time t, in
days is modeled by the logistic growth function.
If 10 fish were in the pond after 15 days,
·Find B
·Find k
·How many fish will be in the tank after 60 days? Round to the nearest whole number.
·F(0) = 6 Initial amount of fish is 6
C = 100 Capacity of the pond is 100
Continued on next screen
Six fish were introduced into a small backyard pond. Because of limited food, space,
and oxygen, the carrying capacity of the pond is 100 fish. The fish population at any
time t, in days is modeled by the logistic growth function.
If 10 fish were in the pond after 15 days,
·B = 47/3
·Find k
·When will the pond contain 60 fish? Round to the nearest whole number.
b. F(15) = 10
Continued on the next screen
Six fish were introduced into a small backyard pond. Because of limited food, space, and oxygen,
the carrying capacity of the pond is 100 fish. The fish population at any time t, in days is modeled by
the logistic growth function.
If 10 fish were in the pond after 15 days,
·B = 47/3
·Find k
c. When will the pond contain 60 fish? Round to the nearest whole number.
c. F(60)