# Lesson 17

Logarithmic Functions

## 17.1: Which One Doesn’t Belong: Functions (5 minutes)

### Warm-up

This warm-up prompts students to compare four equations. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another. A goal of this warm-up is to remind students of the variations in exponential functions they have seen previously, while drawing attention to a new kind of function, logarithmic, which is the focus of the lesson.

### Launch

Arrange students in groups of 2–4. Display the equations for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn't belong.

### Student Facing

Which one doesn’t belong? Be prepared to explain your reasoning.

\(f(x) = 4 \boldcdot (0.75)^x\)

\(g(x) = 4 \boldcdot e^{(0.75x)}\)

\(h(x) = (0.75) \boldcdot 4^x\)

\(j(x) = 4 \boldcdot \log x\)

### Student Response

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### Activity Synthesis

Ask each group to share one reason why a particular item does not belong. 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 one does not belong, attend to students’ explanations and ensure the reasons given are correct.

Conclude the discussion by pointing out that function \(j\) 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: How Long Will It Take? (20 minutes)

### Activity

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).

### Launch

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

*Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct.*Use this routine to give students the opportunity to discuss a common misconception about logarithmic functions. At the appropriate time, display this incorrect and ambiguous response to the first question, “I can rewrite this as \(2^x=h\) because I am trying to find the number of hours for the different values of \(x\).” Ask students to identify the error, critique the reasoning, and write a correct explanation. Invite students to share their critiques and corrected explanations. Listen for and amplify the language students use to explain the meaning of the variables in the correct equation under the given context. This helps students evaluate and improve on the written mathematical arguments of others, as they first encounter a log as a function.

*Design Principle(s): Optimize output (for explanation); Maximize meta-awareness*

### Student Facing

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

- Explain why we can write \(h = \log_{2}{x}\) to represent the number of hours, \(h\), it takes for the one thousand bacteria to reach a population of \(x\) thousand.
- Complete the table with the corresponding values of \(h\).
\(x\) (thousands) 1 2 4 8 16 50 80 \(h\) (hours) - Plot the pairs of values on the coordinate plane. Make two observations about the graph.
- Use the graph to estimate the missing values in the table.
\(x\) (thousands) 10 24 72 \(h\) (hours)

### Student Response

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### Anticipated Misconceptions

Since this is students' first time working with logs as functions, they may be unsure what \(h = \log_2(x)\) means. Invite these students to rewrite the equation in exponential form (\(2^h = x\)). Then ask, in this context, how many hours, \(h\), it takes for the population to reach 2,000? What about 8,000?

### Activity Synthesis

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 \(h = \log_2 x\) 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 we use it to find the time it takes a population that doubles every hour to reach a certain number. Discuss with students:

- “What is the input in this function?” (the population of bacteria)
- “What is the output?” (the time, in hours, that it takes to reach a certain population size)
- “What does the expression \(\log_2 80\) mean? What is its value?” (the number of hours it takes the colony to reach 80 thousand bacteria, which is about 6.32 hours)
- “How did you find the time it takes the colony to reach 72 thousand bacteria?” (use the graph, estimate \(\log_2 72\), or find the value of \(\log_2 72\) using a calculator)

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 \(2^{10} = 1,\!024\). So, \(2^{10}\) thousand is a little more than 1 million. Or, using a base 2 log table or a calculator with \(\log_{2}\)function*, *\(\log_{2}{1,\!000} \approx 9.97\), 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*

## 17.3: Another Logarithmic Function (10 minutes)

### Activity

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 \(10^5\), 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 end behavior of a logarithmic function. Identify students who take this approach so they could share their process during the whole-class discussion.

### Launch

Students may need a reminder about adjusting graphing windows to see function values at large inputs.

### Student Facing

Earlier we saw that \(h = \log_2 x\) represents the number of hours for 1 thousand bacteria, doubling every hour, to reach a population of \(x\), in thousands.

- Suppose the function \(d\), defined by \(d(x) = \log_{10} x\), represents the number of days it takes 1 thousand of another species of bacteria to reach a population of \(x\), in thousands. How is this population of bacteria growing?
- Graph \(d\) using graphing technology. Make two observations about the graph.
- Use your graph to estimate the values of \(d(50)\) and \(d(20,\!000)\). (Adjust your graphing window as needed.) Explain what each value means in this situation.
- Estimate or find the population after 5 days.

### Student Response

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### Student Facing

#### Are you ready for more?

- Without graphing, how do you think the graphs of the equations \(y =\log_2(x)\) and \(y = \log_{10}(x)\) compare? Do they ever meet?
- Graph both equations on the same axes to test your conjectures.

### Student Response

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### Activity Synthesis

Make sure students see that this bacteria population is growing by a factor of 10 every day. Highlight that here the input (population) increases tenfold before the output (days) increases by 1. Relate this observation to the meaning of base 10 logarithm.

Focus the discussion on students’ observations and interpretations of the graph, and how the graphs representing \(\log_{10} x\) and \(\log_2 x\) compare.

Invite previously identified students who use a graph to answer the last question to share their approach. If no one took this path, ask students to consider if it is a helpful strategy. Students may be able to easily see on the graph the input value that gives an output of 4, but to use the graph to find out what input value gives an output of 5 requires looking into very large input values. A key takeaway here is that the input has to increase by a factor of 10 for the output to increase by 1.

*Speaking: MLR8 Discussion Supports.*Use this routine to support whole-class discussion. To help students share their observations about the graph and how the graphs of \(\log_{2}x\) and \(\log_{10}x\) compare, provide sentence frames such as “I notice that . . .”, “It looks like _____ represents . . .”, “_____ and _____ are alike because . . .”, “_____ and _____ are different because . . . .”

*Design Principle(s): Optimize output (for comparison); Support sense-making*

*Action and Expression: Internalize Executive Functions.*Provide students with a Venn diagram to organize their observations about how the graphs of \(\log_{10} x\) and \(\log_{2} x\) compare. Demonstrate labeling each circle with its own expression. Invite students to write similarities in the center overlapping areas (shape of the graph) and differences in the right and left areas according to each expression (one doubles, the other gets multiplied by 10). For additional support, scribe the student responses on a display to share with the whole class.

*Supports accessibility for: Language; Organization*

## Lesson Synthesis

### Lesson Synthesis

Invite students to consider the two log functions they saw in the lesson. To clarify the connections between the two functions, and between log functions and exponential functions, discuss questions such as:

- “How are the two log functions alike?” (They both represent the elapsed time as a function of population and are expressed using a logarithm. They have very similar graphs.)
- “How are they different?” (The bases are different. The first represented a population growing by a factor of 2 over time, while the second represented a population growing by a factor of 10 over time.)
- “When we evaluate an expression such as \(\log_2 x\) for some value of \(x\), what are we looking for?” (We are looking for the exponent by which to raise a base 2 to produce \(x\). In this case, the \(x\) 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 \(x\), 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.

## 17.4: Cool-down - Which Graph Represents Which Function? (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

Earlier we have 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** 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 \(P = 1 \boldcdot 2^d\) or \(P = 2^d\) to represent the population, in thousands, after \(d\) 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 \(\displaystyle d = \log_2 P\). In this function, the input is \(P\), population in thousands, and the output is \(d\), time in decades. Here is a graph representing that function.

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?” In this case, the answer is about 10 decades, because one million is 1,000 thousands and \(\log_2 1,\!000 \approx 10\) (or, thinking in terms of powers of 2, we know that \(2^{10} =1,\!024\)).

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 \(\displaystyle d = \log_{10} P\).

From the graph, we can see that it takes only 3 decades to reach 1,000 thousands, because \(\log_{10} 1,\!000=3\) (or \(10^3 = 1,\!000\)).