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Unit 5, Lesson 17

Logarithmic Functions

Algebra 2

17.1

Warm-up

5 mins

Which Three Go Together: Functions

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Which Three Go Together?

Materials

None

Activity Narrative

This Warm-up prompts students to compare four functions. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk, in comparison to one another, about characteristics of the items.

Launch

Arrange students in groups of 2–4. Display the functions. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three functions that go together and can explain why. Next, tell students to share their response with their group and then to work together to find as many sets of three as they can.

Activity

None

Student Task Statement

Which three go together? Why do they go together?

Activity Synthesis

Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure that the reasons given are correct.

During the discussion, prompt students to explain the meaning of any function terminology they use, such as “increasing,” “exponential function,” and “base,” and to clarify their reasoning as needed. Consider asking:

“How do you know _____?”
“What do you mean by _____?”
“Can you say that in another way?”

Conclude the discussion by pointing out that function is called a logarithmic function or a log function. Explain that we will look at some functions during today's lesson where the output is the logarithm of the input.

17.2

Activity

20 mins

How Long Will It Take?

Instructional Routines

Graph It MLR3: Critique, Correct, Clarify

Materials

None

Activity Narrative

This activity introduces students to a logarithmic function in the context of population growth. Prior to this point, students have written exponential functions to represent populations growing or shrinking over time. Here students use a logarithmic expression to represent elapsed time as a function of population size.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

This activity uses the Critique, Correct, Clarify math language routine to advance representing and conversing as students critique and revise mathematical arguments.

Launch

If needed, prompt students to recall the meaning of log statements in different bases. For example, ask, “What does mean? ?” ( is the exponent by which we raise a base of 2 to get 10. That is, . is the exponent by which we raise a base of 5 to get 60.)

Read the problem stem and first instruction together. Then, use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to the first question by correcting errors, clarifying meaning, and adding details.

Display this first draft:
“I can rewrite the equation as because I am trying to find the number of hours for the different values of .”
Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
Give students 2–4 minutes to work with a partner to revise the first draft.
Select 1–2 students or groups to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

17.3

Activity

10 mins

Another Logarithmic Function

Instructional Routines

Graph It MLR8: Discussion Supports

Materials

To Gather

Graphing technology

Activity Narrative

In this activity, students encounter another logarithmic function in a different base but grounded in the same context. They interpret the base-10 logarithm and graph the function. Then they use the graph to estimate the function value at different inputs and, along the way, interpret what the values tell us about the situation (MP2).

To find the population after 5 days (the last question), students may simply write , but some may try to use the graph to estimate the input value, given 5 as the output. If using the graph, students will notice that they have to keep expanding the graphing window or the domain to determine the input when the output is 5. This is an opportunity for thinking about the end behavior of a logarithmic function. Identify students who take this approach so they could share their process during the whole-class discussion.

Lesson Synthesis

The purpose of this discussion is for students to recognize features of logarithmic function graphs.

Display the graphs of and on the same coordinate plane.

<p>Graph of two equations on an axis.</p>

Tell students that these are the graphs of and . Then ask students,

“Which graph matches with which equation? Explain your reasoning.” ( is the function on top for values of after 1. I know because it has the point on it, and .)
“How are the two log functions and graphs alike?” (They both represent the elapsed time as a function of population and are expressed using a logarithm. They both go through the point . They both start increasing very steeply, then more slowly as increases.)
“How are they different?” (The bases are different. The function represents a population growing by a factor of 2 over time, while the other function represents a population growing by a factor of 10 over time. The graph of increases more quickly than the other graph does for small -values, but then levels out more quickly.)
“When we evaluate an expression such as for some value of , how can you find the output?” (The output is the exponent for the base 2 that produces . In this situation, the is a certain population size, and the exponent is time in hours.)

Highlight the end behavior of the graph representing a logarithmic function. The graph becomes increasingly horizontal for increasingly larger values of , which tells us that the input has to increase by a lot before the output grows by 1. For the base-2 logarithm, the input has to increase by a factor of 2. For the base-10 logarithm, it has to increase by a factor of 10. Eventually, though, both graphs can have -values as large as you could want.

to access Cool-down.

Student Lesson Summary

Earlier, we studied exponential relationships where the input values are the exponent in the function. Sometimes we want to express an exponential relationship where the values we want to find, the outputs, are the exponents. A logarithmic function, or a function of the form , where is a constant and is a positive number, can help us do that.

For example: Suppose the population of a town starts at one thousand and doubles every decade since first measured. We can write (or ) to represent the population, in thousands, after decades.

But if we want to know how long, in decades, it would take to reach certain population sizes, in thousands, we can write a logarithmic function . In this function, the input is , population in thousands, and the output is , time in decades. Here is a graph representing that function.

<p>Coordinate plane, P, population in thousands, d, time in decades. Curve d = log base 2 of p is graphed through 1 comma 0, 256 thousand comma 8 , approximately 1000 thousand comma 9 point 9 7.</p>

We can use the graph to estimate the answer to a question such as, “How many decades will it take for the population to reach a million?” by finding the value 1,000 (1,000 thousands is 1 million) on the horizontal axis and following that up to see that the graph has a point near . This means that the population will reach 1 million after about 10 decades. We can check the answer by finding that .Suppose the population of that town expands by a factor of 10 every decade instead of by a factor of 2. The function representing the time it takes to reach a certain population, in thousands, would be .

<p>Coordinate plane, P, population in thousands, d, time in decades. Curve d = log base 10 of p is graphed through 1 comma 0, 100 thousand comma 2 , 1000 thousand comma 3.</p>

From the graph, we can see that it takes only 3 decades to reach 1,000 thousands, because the point appears to be on the graph (and ).

Student Task Statement

A colony of 1,000 bacteria doubles in population every hour.

Explain why we can write to represent the number of hours, , it takes for the one thousand bacteria to reach a population of thousand.
Complete the table with the corresponding values of .

(thousands) 1 2 4 8 16 50 80

(hours)
Plot the pairs of values on the coordinate plane. Then sketch a curve that connects the points. Make two observations about the graph.
Use the graph to estimate the missing values in the table.

(thousands) 10 24 72

(hours)

(thousands)	1	2	4	8	16	50	80
(hours)

(thousands)	10	24	72
(hours)

Activity Synthesis

The purpose of the discussion is for students to become familiar with the graph of a logarithmic function. Display the graph of .

Focus the discussion on the graph of the function and students’ observations of its features. Ask them to describe how the graph is different from graphs representing exponential, linear, quadratic, or other functions they have seen.

Introduce as a logarithmic function. In this case, the base of the logarithm is 2, which tells us it is related to a quantity that changes by a factor of 2. Here, it is used to find the time it takes a population that doubles every hour to reach a certain number. Discuss with students:

“What does the input of this function mean in this situation?” (the population of bacteria)
“What does the output mean?” (the time, in hours, that it takes to reach a certain population size)
“What does the expression mean? What is its value?” (the number of hours it takes the colony to reach 24 thousand bacteria, which is about 4.6 hours)
“How did you find the time it takes the colony to reach 72 thousand bacteria?” (I looked on the graph I sketched to estimate to find where a vertical line at would meet the curve. It looks to be a little more than 6, so I estimated 6.2 hours.)

Point out that, like graphs of other functions, log graphs can be used to answer questions about the relationship, though not always precisely. We can, however, use the equation defining a log function to solve problems with exact answers.

If time allows, conclude the discussion by asking, “About how many hours will it take for the population to reach 1 million?” (It would take about 10 hours because one million bacteria equals 1,000 thousand bacteria and . So, thousand is a little more than 1 million. Or, using a base-2 log table or a calculator with function, , which is about 10.)

Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of logarithmic functions. Using the example from the activity, label the parts of the equation with quantities it represents and the meanings of the numbers and variables used in the example.
Supports accessibility for: Conceptual Processing, Language

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Lesson 17

Logarithmic Functions

Warm-up

Which Three Go Together: Functions

Standards Alignment

Building On

HSA-SSE.A.1.a

Addressing

Building Toward

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Activity Synthesis

Activity

How Long Will It Take?

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Another Logarithmic Function

Instructional Routines

Materials

To Gather

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Are You Ready for More?

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Standards Alignment

Building On

Addressing

HSF-IF.C.7.e

HSF-LE.A.4

Building Toward

Standards Alignment

Building On

Addressing

HSF-IF.C.7.e

HSF-LE.A.4

Building Toward