Lesson 17

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.

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.

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.

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).

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.)

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? 

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.)

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.

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

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.