NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 21 PRECALCULUS AND ADVANCED TOPICS
Lesson 21: Logarithmic and Exponential Problem Solving
350
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 21: Logarithmic and Exponential Problem Solving
Student Outcomes
Students understand the inverse relationship between logarithms and exponents and apply their
understanding to solve real-world problems.
Lesson Notes
This lesson uses the context of radiocarbon dating to understand and apply the inverse relationship between logarithms
and exponents when solving problems. A quick Internet search reveals several news stories about recent discoveries of
woolly mammoth remains. Radiocarbon dating is one of several methods used by archaeologists and anthropologists to
date their findings. Students have studied real-world situations that are modeled by exponential functions since
Algebra I. In Algebra II, they were able to develop more precise solutions to modeling and application problems
involving exponential functions because they learned to use logarithms to solve equations analytically (F-BF.B.4,
F.LE.A.4). Students also learned about situations that could be modeled by logarithmic functions. In this lesson,
students use the inverse relationship between logarithms and exponents as a basis for making sense of and solving real-
world problems. Throughout the lesson, students create models, compute using models, and interpret the results
(MP.4).
Classwork
Opening (5 minutes)
Have students read the Opening to themselves and jot down one question that they have about the reading. Have them
share and discuss this question with a partner. If time permits, students can research the answers to any questions that
the teacher or others in the class cannot answer. The following Web-based resources can help the teacher and students
learn more about radiocarbon dating and woolly mammoth discoveries.
http://en.wikipedia.org/wiki/Radiocarbon_dating
http://science.howstuffworks.com/environmental/earth/geology/carbon-14.htm
http://en.wikipedia.org/wiki/Woolly_mammoth
http://ngm.nationalgeographic.com/2009/05/mammoths/mueller-text
After a brief whole-class discussion, move students on to the Exploratory Challenge.
Woolly mammoths, elephant-like mammals, have been extinct for thousands of years. In the last decade, several well-
preserved woolly mammoths have been discovered in the permafrost and icy regions of Siberia. Using a technique called
radiocarbon (Carbon-14) dating, scientists have determined that some of these mammoths died nearly ππ, πππ years
ago.
This technique was introduced in 1949 by the American chemist Willard Libby and is one of the most important tools
archaeologists use for dating artifacts that are less than ππ, πππ years old. Carbon-14 is a radioactive isotope present in
all organic matter. Carbon-14 is absorbed in small amounts by all living things. The ratio of the amount of normal carbon
(Carbon-12) to the amount of Carbon-14 in all living organisms remains nearly constant until the organism dies. Then, the
Carbon-14 begins to decay because it is radioactive.
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 21 PRECALCULUS AND ADVANCED TOPICS
Lesson 21: Logarithmic and Exponential Problem Solving
351
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Exploratory Challenge/Exercises 1β14 (20 minutes)
Organize students into small groups, and give them about 10 minutes to work these exercises. The goal is for them to
create an exponential function to model the data. Monitor groups to make sure they are completing the table entries
correctly. There is quite a bit of number sense and quantitative reasoning required in these exercises.
Students may wish to construct the graphs on graph paper. Pay attention to how they scale the graphs. Graphing
calculators or other graphing technology can also be used to construct the scatter plots and graphs of the functions.
Exploratory Challenge/Exercises 1β14
By examining the amount of Carbon-14 that remains in an organism after death, one can determine its age. The half-life
of Carbon-14 is π, πππ years, meaning that the amount of Carbon-14 present is reduced by a factor of π
π every π, πππ
years.
1. Complete the table.
Years Since Death π π, πππ ππ, πππ ππ, πππ ππ, πππ ππ, πππ ππ, πππ ππ, πππ ππ, πππ
C-14 Atoms Remaining
Per π. π Γ πππ
C-12 Atoms
ππ, πππ π, πππ π, πππ π, πππ πππ πππ. π πππ. ππ ππ. πππ ππ. πβπππ
Let πͺ be the function that represents the number of C-14 atoms remaining per π. π Γ πππ C-12 atoms π years after death.
2. What is πͺ(ππ, πππ)? What does it mean in this situation?
πͺ(πππππ) = ππππ
Each time the years since death increase by π, πππ, you have to reduce the C-14 atoms per π. π Γ πππ C-12 atoms by
a factor of π
π. After the organism has been dead for ππ, πππ years, the organism contains π, πππ C-14 atoms per
π. π Γ πππ C-12 atoms.
3. Estimate the number of C-14 atoms per π. π Γ πππ C-12 atoms you would expect to remain
in an organism that died ππ, πππ years ago.
It would be slightly more than π, πππ atoms, so approximately π, πππ.
4. What is πͺβπ(πππ)? What does it represent in this situation?
πͺβπ(πππ) = ππ, πππ This represents the number of years since death when there are πππ C-14 atoms per π. π Γ πππ C-12 atoms remaining in the sample.
5. Suppose the ratio of C-14 to C-12 atoms in a recently discovered woolly mammoth was
found to be π. πππβπππ. Estimate how long ago this animal died.
We need to write this as a ratio with a denominator equal to π. π Γ πππ.
π. πππβπππ = π. π Γ ππβπ =π. π Γ πππ
π. π Γ πππ
So, if the number of C-12 atoms is π. π Γ πππ, then the number of C-14 atoms would be πππ. This animal would
have died between ππ, πππ and ππ, πππ years ago.
MP.3 Scaffolding:
Let students use a calculator to do the heavy computational lifting. Students could also model the table information using a spreadsheet. The regression features could be used to generate an exponential model for this situation as well.
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 21 PRECALCULUS AND ADVANCED TOPICS
Lesson 21: Logarithmic and Exponential Problem Solving
352
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6. Explain why the πͺβπ(πππ) represents the answer to Exercise 5.
πͺβπ(πππ) means the value of π when πͺ(π) = πππ. We wanted the time when there would be πππ C-14 atoms for
every π. π Γ πππ C-12 atoms.
7. What type of function best models the data in the table you created in Exercise 1? Explain your reasoning.
Since the data is being multiplied by a constant factor each time the years increase by the same amount, this data
would be modeled best by an exponential function.
Exercises 8 and 9 provide an opportunity for the teacher to check for student understanding of essential prerequisite
skills related to writing and graphing exponential functions. In both Algebra I and Algebra II, students have modeled
exponential functions when given a table of values. The challenge for students is to see how well they deal with the half-
life parameter. Students may choose to use a calculator to do an exponential regression equation as well. Then, in
Exercise 10, the teacher can assess how well students understand that logarithmic and exponential functions are
inverses. They create the graph of the inverse of πΆ by exchanging the π‘ and πΆ(π‘) coordinates and plotting the points.
The result of their work in Lesson 20 is that they should now understand that the inverse of any exponential function of
the form π¦ = π β πππ₯ is a logarithmic function with base π. Use the questions below to provide additional support as
groups are working.
What type of function makes sense to use to model πΆ?
An exponential function because whenever the time increased by a consistent amount, the value of πΆ
was multiplied by 1
2.
How can you justify your choice of function using the graph in Exercise 9?
I can see that every 5,730 years, the amount of πΆ is reduced by a factor of 1
2. The graph is decreasing
but never reaches 0, so it appears to look like a model for exponential decay.
What is the base of the function? What is the π¦-intercept?
The base is 1
2. The π¦-intercept would be the amount when π‘ = 0.
How does the half-life parameter affect how you write the formula for πΆ in Exercise 8?
Since the years are not increasing by 1 but by 5,730, we need to divide the time variable by 5,730 in
our exponential function.
How do you know that the graph of the function in Exercise 10 is the inverse of the graph of the function in
Exercise 9?
Because we exchanged the domain and range values from the table to create the graph of the inverse.
What type of function is the inverse of an exponential function?
In the last lesson, we learned that the inverse of an exponential function is a logarithmic function.
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 21 PRECALCULUS AND ADVANCED TOPICS
Lesson 21: Logarithmic and Exponential Problem Solving
353
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
8. Write a formula for πͺ in terms of π. Explain the meaning of any parameters in your formula.
πͺ(π) = πππππ(π
π)
πππππ
The ππ, πππ represents the number of C-14 atoms per π. π Γ πππ C-12 atoms present at the death of the organism.
The π, πππ and the π
π indicate that the amount is halved every π, πππ years.
9. Graph the set of points (π, πͺ(π)) from the table and the function πͺ to verify that your formula is correct.
10. Graph the set of points (πͺ(π), π) from the table. Draw a smooth curve connecting those points. What type of
function would best model this data? Explain your reasoning.
This data would best be modeled by a logarithmic function since the data points represent points on the graph of the
inverse of an exponential function.
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 21 PRECALCULUS AND ADVANCED TOPICS
Lesson 21: Logarithmic and Exponential Problem Solving
354
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
11. Write a formula that gives the years since death as a function of the amount of C-14 remaining per π. π Γ πππ
C-12 atoms.
Take the equation πͺ(π) = πππππ (ππ), and use the variables π and π in place of π and πͺ(π). Find the inverse by
exchanging the π and π variables and solving for π.
π = πππππ(π
π)
πππππ
π = πππππ(π
π)
πππππ
π
πππππ= (π
π)
πππππ
π₯π¨π π(
π
πππππ) = π₯π¨π
π(π
π)
πππππ
π₯π¨π π(
π
πππππ) = π₯π¨π
2(πβπ)
πππππ
π₯π¨π π(
π
πππππ) =
π
πππππ₯π¨π
π(πβπ)
π₯π¨π π(
π
πππππ) = β
π
ππππ
π = βππππ π₯π¨π π (π
πππππ)
π(π) = βππππ π₯π¨π π (π
πππππ)
In this formula, π is the number of C-14 atoms for every π. π Γ πππ C-12 atoms, and π(π) is the time since death.
12. Use the formulas you have created to accurately calculate the following:
a. The amount of C-14 atoms per π. π Γ πππ C-12 atoms remaining in a sample after ππ, πππ years
π = πππππ(π
π)
πππππππππ
πͺ(πππππ) β ππππ
There are approximately π, πππ C-14 atoms per π. π Γ πππ C-12 atoms in a sample that died ππ, πππ years
ago.
b. The years since death of a sample that contains πππ C-14 atoms per π. π Γ πππ C-12 atoms
π(π) = βππππ ππππ (πππ
πππππ)
π(πππ) β πππππ
An organism containing πππ C-14 atoms per π. π Γ πππ C-12 atoms died ππ, πππ years ago.
MP.4
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 21 PRECALCULUS AND ADVANCED TOPICS
Lesson 21: Logarithmic and Exponential Problem Solving
355
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
c. πͺ(ππ, πππ)
π = πππππ(π
π)
πππππππππ
πͺ(πππππ) β πππ
After an organism has been dead for ππ, πππ years, there are approximately πππ C-14 atoms per
π. π Γ πππ C-12 atoms.
d. πͺβπ(π, πππ)
π(π) = βππππ π₯π¨π π (ππππ
πππππ)
To find this amount, evaluate π(ππππ) β πππππ.
When there are π, πππ C-14 atoms per π. π Γ πππ C-12 atoms, the organism has been dead for ππ, πππ years.
13. A baby woolly mammoth that was discovered in ππππ died approximately ππ, πππ years ago. How many C-14
atoms per π. π Γ πππ C-12 atoms would have been present in the tissues of this animal when it was discovered?
Evaluate πͺ(πππππ) β ππ atoms of C-14 per ππ Γ πππ atoms of C-12.
14. A recently discovered woolly mammoth sample was found to have a red liquid believed to be blood inside when it
was cut out of the ice. Suppose the amount of C-14 in a sample of the creatureβs blood contained π, πππ atoms of
C-14 per π. π Γ πππ atoms of C-12. How old was this woolly mammoth?
Evaluate πͺβπ(ππππ) β ππππ. The woolly mammoth died approximately ππ, πππ years ago.
Have students present different portions of this Exploratory Challenge, and then lead a short discussion to make sure all
students have been able to understand the inverse relationship between logarithmic and exponential functions.
Discuss different approaches to finding the logarithmic function that represented the inverse of πΆ. One of the challenges
for students may be dealing with the notation. When applying inverse relationships in real-world situations, be sure to
explain the meaning of the variables.
Exercises 15β18 (10 minutes)
In Exercises 15β18, students generalize the work they did in earlier exercises. Because the half-life of C-14 is 5,730
years, the carbon-dating technique only produces valid results for samples up to 50,000 years old. For older samples,
scientists can use other radioactive isotopes to date the rock surrounding a fossil and infer the fossilβs age from the age
of the rock.
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 21 PRECALCULUS AND ADVANCED TOPICS
Lesson 21: Logarithmic and Exponential Problem Solving
356
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Exercises 15β18
Scientists can infer the age of fossils that are older than ππ, πππ years by using similar dating techniques with other
radioactive isotopes. Scientists use radioactive isotopes with half-lives even longer than Carbon-14 to date the
surrounding rock in which the fossil is embedded.
A general formula for the amount π¨ of a radioactive isotope that remains after π years is π¨ = π¨π (ππ)
ππ
where π¨π is the amount of radioactive substance present initially and π is the half-life of the radioactive substance.
15. Solve this equation for π to find a formula that infers the age of a fossil by dating the age of the surrounding rocks.
π¨ = π¨π (π
π)
ππ
π¨
π¨π= (π
π)
ππ
π¨
π¨π= π
βππ
π₯π¨π π (π¨
π¨π) = β
π
π
π = βπ π₯π¨π π (π¨
π¨π)
16. Let (π) = π¨π (ππ)
ππ
. What is π¨βπ(π)?
π¨βπ(π) = βπ πππ2 (π
π¨π)
17. Verify that π¨ and π¨βπ are inverses by showing that π¨(π¨βπ(π)) = π and π¨βπ(π¨(π)) = π.
π¨(π¨βπ(π)) = π¨π (π
π)
βππ₯π¨π π(ππ¨π
)
π
= π¨π(πβπ)βπ₯π¨π π(
ππ¨π
)
= π¨π(π)π₯π¨π π(
ππ¨π
)
= π¨π (π
π¨π)
= π
And
π¨βπ(π¨(π)) = βπ π₯π¨π π
(
π¨π (
ππ)
ππ
π¨π
)
= βπ π₯π¨π 2 (π
π)
ππ
= βπ βπ
πβ π₯π¨π π (
π
π)
= βπ β βπ
= π
Scaffolding:
For students who need a more concrete approach, select particular values for π΄0 and β, and have students solve the equation for π‘.
Do this a few times, and then have students try to generalize the solution steps.
Consider also using
π΄ = π΄0(2)βπ‘
β instead of 1
2
so that students quickly see the connection to base 2 logarithms.
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 21 PRECALCULUS AND ADVANCED TOPICS
Lesson 21: Logarithmic and Exponential Problem Solving
357
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
18. Explain why, when determining the age of organic materials, archaeologists and anthropologists would prefer to use
the logarithmic function to relate the amount of a radioactive isotope present in a sample and the time since its
death.
Defining the years since death as a function of the amount of radioactive isotope makes more sense since
archaeologists and anthropologists are trying to determine the number of years since the death of an organism from
a sample. They know the amount of the radioactive isotope and can use that as an input into the formula to
generate the number of years since its death.
Closing (5 minutes)
Ask students to respond to the questions below with a partner or individually in writing.
Which properties of logarithms and exponents are most helpful when verifying that these types of functions
are inverses?
The definition of logarithm, which states that if π₯ = ππ¦, then π¦ = logπ(π₯), and the identities,
logπ(ππ₯) = π₯ and πlogπ(π₯) = π₯
What is the inverse function of π(π₯) = 3π₯? What is the inverse of π(π₯) = log(π₯)?
πβ1(π₯) = log3(π₯), and πβ1(π₯) = 10π₯
Suppose a function π has a domain that represents time in years and a range that represents the number of
bacteria. What would the domain and range of πβ1 represent?
The domain would be the number of bacteria, and the range would be the time in years.
Exit Ticket (5 minutes)
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 21 PRECALCULUS AND ADVANCED TOPICS
Lesson 21: Logarithmic and Exponential Problem Solving
358
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Name Date
Lesson 21: Logarithmic and Exponential Problem Solving
Exit Ticket
Darrin drank a latte with 205 milligrams (mg) of caffeine. Each hour, the caffeine in Darrinβs body diminishes by about
8%.
a. Write a formula to model the amount of caffeine remaining in Darrinβs system after each hour.
b. Write a formula to model the number of hours since Darrin drank his latte based on the amount of caffeine in
Darrinβs system.
c. Use your equation in part (b) to find how long it takes for the caffeine in Darrinβs system to drop below
50 mg.
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 21 PRECALCULUS AND ADVANCED TOPICS
Lesson 21: Logarithmic and Exponential Problem Solving
359
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Exit Ticket Sample Solutions
Darrin drank a latte with πππ milligrams (mg) of caffeine. Each hour, the caffeine in Darrinβs body diminishes by about
π%.
a. Write a formula to model the amount of caffeine remaining in Darrinβs system after each hour.
π(π) = πππ β (π β π%)π π(π) = πππ β (π. ππ)π
b. Write a formula to model the number of hours since Darrin drank his latte based on the amount of caffeine in
Darrinβs system.
π = πππ(π. ππ)π π
πππ= π. πππ
π₯π§ (π
πππ) = π₯π§(π. ππ)π
π₯π§ (π
πππ) = π β π₯π§(π. ππ)
π =π₯π§ (
ππππ
)
π₯π§ (π. ππ)
Thus,
π(π) =π₯π§(
ππππ
)
π₯π§ (π. ππ).
Alternatively,
π = πππ(π. ππ)π π
πππ= π. πππ
π₯π¨π π.ππ (π
πππ) = π.
And by the change of base property,
π(π) =π₯π§ (
ππππ
)
π₯π§(π. ππ).
c. Use your equation in part (b) to find how long it takes for the caffeine in Darrinβs system to drop below
ππ π¦π .
π =π₯π§ (
πππππ
)
π₯π§ (π. ππ)
It would take approximately ππ. πππ hours for the caffeine to drop to ππ π¦π ; therefore, it would take
approximately ππ hours for the caffeine to drop below ππ π¦π .
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 21 PRECALCULUS AND ADVANCED TOPICS
Lesson 21: Logarithmic and Exponential Problem Solving
360
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Problem Set Sample Solutions
1. A particular bank offers π% interest per year compounded monthly. Timothy wishes to deposit $π, πππ.
a. What is the interest rate per month?
π. ππ
ππ= π. πππ
b. Write a formula for the amount π¨ Timothy has after π months.
π¨ = ππππ(π.πππ)π
c. Write a formula for the number of months it takes Timothy to have π¨ dollars.
π =π₯π§(
π¨ππππ
)
π₯π§(π. πππ)
d. Doubling-time is the amount of time it takes for an investment to double. What is the doubling-time of
Timothyβs investment?
π =π₯π§(π)
π₯π§(π. πππ)
β πππ. ππ
It takes πππ months for Timothyβs investment to double.
e. In general, what is the doubling-time of an investment with an interest rate of π
ππ per month?
π =π₯π§(π)
π₯π§ (π +πππ)
2. A study done from 1950 through 2000 estimated that the world population increased on average by π. ππ% each
year. In 1950, the world population was π, πππ million.
a. Write a formula for the world population π years after 1950. Use π to represent the world population.
π = ππππ(π.ππππ)π
b. Write a formula for the number of years it takes to reach a population of π.
π =π₯π§ (
πππππ
)
π₯π§(π. ππππ)
c. Use your equation in part (b) to find when the model predicts that the world population is ππ billion.
π =π₯π§ (
πππππππππ
)
π₯π§(π. ππππ)
β ππ. πππ
According to the model, it takes about ππππ
years from 1950 for the world population to reach ππ billion; this
would be in 2028.
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 21 PRECALCULUS AND ADVANCED TOPICS
Lesson 21: Logarithmic and Exponential Problem Solving
361
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
3. Consider the case of a bank offering π (given as a decimal) interest per year compounded monthly, if you
deposit $π·.
a. What is the interest rate per month?
π
ππ
b. Write a formula for the amount π¨ you have after π months.
π¨ = π·(π +π
ππ)π
c. Write a formula for the number of months it takes to have π¨ dollars.
π₯π§ (π¨
π·) = π β π₯π§ (π +
π
ππ)
π =π₯π§ (
π¨π·)
π₯π§ (π +πππ)
d. What is the doubling-time of an investment earning π% interest per year, compounded monthly? Round up
to the next month.
π = (π +π. ππ
ππ)π
π₯π§(π) = π β π₯π§ (π +π. ππ
ππ)
π =π₯π§(π)
π₯π§ (π +π. ππππ
)β πππ. ππ
It would take πππ months or ππ years in order to double an investment earning π% interest per year,
compounded monthly.
4. A half-life is the amount of time it takes for a radioactive substance to decay by half. In general, we can use the
equation π¨ = π·(ππ)π
for the amount of the substance remaining after π half-lives.
a. What does π· represent in this context?
The initial amount of the radioactive substance
b. If a half-life is ππ hours, rewrite the equation to give the amount after π hours.
π =π
ππ
π¨ = π·(π
π)
πππ
c. Use the natural logarithm to express the original equation as having base π.
π¨ = π·ππ₯π§(ππ)π
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 21 PRECALCULUS AND ADVANCED TOPICS
Lesson 21: Logarithmic and Exponential Problem Solving
362
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
d. The formula you wrote in part (c) is frequently referred to as the βPertβ formula, that is, π·πππ. Analyze the
value you have in place for π in part (c). What do you notice? In general, what do you think π represents?
π = π₯π§ (π
π)
It seems like the π value always represents the natural logarithm of the growth rate per π. If π is a number of
half-lives, then π is the natural logarithm of π
π.
e. Jess claims that any exponential function can be written with base π; is she correct? Explain why.
She is correct. No matter what the original rate is, say π, we can always rewrite the rate as πππ(π), so π is
always a possible base for an exponential function. Similarly, we could always rewrite logarithms in terms of
the natural logarithm.
5. If caffeine reduces by about ππ% per hour, how many hours π‘ does it take for the amount of caffeine in a body to
reduce by half (round up to the next hour)?
π
π= π β (π. π)π
π =π₯π§(
ππ)
π₯π§(π. π)
β π.ππππ
It takes about π hours for the caffeine to reduce by half.
6. Iodine-123 has a half-life of about ππ hours, emits gamma-radiation, and is readily absorbed by the thyroid.
Because of these facts, it is regularly used in nuclear imaging.
a. Write a formula that gives you the percent π of iodine-123 left after π half-lives.
π¨ = π·(π
π)π
π¨
π·= (π
π)π
π = (π
π)π
b. What is the decay rate per hour of iodine-123? Approximate to the nearest millionth.
π =π
ππ
π = (π
π)
πππ
π = ((π
π)π/ππ
)
π
π β (π. πππβπππ)π
Iodine-123 decays by about π. πππβπππ per hour, or π. ππππ%.
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 21 PRECALCULUS AND ADVANCED TOPICS
Lesson 21: Logarithmic and Exponential Problem Solving
363
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
c. Use your result to part (b). How many hours π would it take for you to have less than π% of an initial dose of
iodine-123 in your system? Round your answer to the nearest tenth of an hour.
π. ππ = (π. πππβπππ)π
π =π₯π§(π. ππ)
π₯π§(π. πππβπππ)
β ππ. π
It would take approximately ππ. π hours for you to have less than π% of an initial does of iodine-123 in your
system.
7. An object heated to a temperature of ππΒ°π is placed in a room with a constant temperature of ππΒ°π to cool down.
The objectβs temperature π» after π minutes can be given by the function π»(π) = ππ + πππβπ.πππβππππ.
a. How long does it take for the object to cool down to ππΒ°π?
ππ = ππ + πππβπ.πππβππππ
π
π= πβπ.πππβππππ
π =π₯π§(
ππ)
βπ. πππβπππβ ππ. ππππ
About ππ minutes
b. Does it take longer for the object to cool from ππΒ°π to ππΒ°π or from ππΒ°π to ππ. πΒ°π?
Since it is an exponential decay function, it takes longer for the object to cool from ππΒ°π to ππ. πΒ°π than it
takes for the object to cool from ππΒ°π to ππΒ°π. The function levels off as it approaches ππΒ°π, so it takes
progressively longer. It would take an additional ππ minutes to cool down to ππ. πΒ°π after getting to ππΒ°π.
c. Will the object ever be ππΒ°π if kept in this room?
It will effectively be ππΒ°π eventually, but mathematically it will never get there. After πππ minutes, the
temperature will be about ππ. ππππΒ°π.
d. What is the domain of π»βπ? What does this represent?
The domain of the inverse represents the possible temperatures that the object could be, so (ππ, ππ].
8. The percent of usage of the word judgment in books can be modeled with an exponential decay curve. Let π· be the
percent as a function of π, and let π be the number of years after 1900; then, π·(π) = π. πππβπππβπ β πβπ.πππβπππβππ.
a. According to the model, in what year was the usage π. π% of books?
π·βπ(π) =π₯π§ (
ππ. πππβπππβπ
)
βπ. πππβπππβπ
According to the inverse of the model, we get a value of βπππ, which corresponds to the year ππππ.
b. When does the usage of the word judgment drop below π. πππ% of books? This model was made with data
from 1950 to 2005. Do you believe your answer is accurate? Explain.
We get a value of πππ, which corresponds to the year π, πππ. It is unlikely that the model would hold up well
in either part (a) or part (b) because these years are so far in the past and future.
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 21 PRECALCULUS AND ADVANCED TOPICS
Lesson 21: Logarithmic and Exponential Problem Solving
364
This work is derived from Eureka Math β’ and licensed by Great Minds. Β©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
c. Find π·βπ. What does the domain represent? What does the range represent?
π = π. πππβπππβπ β πβπ.πππβπππβππ
βπ.πβππβπππβππ = π₯π§(π
π. πππβπππβπ)
π =π₯π§(
ππ. πππβπππβπ
)
βπ. πππβπππβπ
The domain is the percent of books containing the word βjudgmentβ while the range is the number of years
after π, πππ.
Top Related