Lesson 15

Using Graphs and Logarithms to Solve Problems (Part 1)

15.1: Using a Graph to Estimate (5 minutes)

Warm-up

This warm-up is students’ first opportunity to make an explicit connection between the graph of an exponential function with base \(e\) and the natural logarithm. Students are prompted to explain why the graph representing an exponential function can be used to estimate a natural logarithm (MP3) and then to make an estimate. They also practice using a calculator to evaluate a natural logarithm.

As students discuss, monitor for those who can articulate the connections between the given graph and logarithm. Invite them to share during the whole-class discussion.

Launch

Arrange students in groups of 2. Give them a moment of quiet time to think about the questions. Then, ask them to discuss their thinking with a partner. Follow with a whole-class discussion. 

Student Facing

Coordinate plane, x, 0 to 6 by 1, y, 0 to 250 by 50. Exponential function f through 0 comma 1, and approximate points 2 comma 2 point 7, 3 comma 20 point 1, 4 comma 54 point 6, 5 comma 148 point 4.

Here is a graph that represents an exponential function with base \(e\), defined by \(f(x) = e^x\).

  1. Explain how to use the graph to estimate logarithms such as \(\ln 100\).
  2. Use the graph to estimate \(\ln 100\).
  3. How can you use a calculator to check your estimate? What would you enter into the calculator?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Focus the discussion on the first question. Invite previously identified students to share their explanations. An important takeaway for students is seeing that \(\ln 100\) is the exponent to which we raise a base \(e\) to produce 100. The graph shows the value of the function \(e^x\), so we can trace the graph and estimate the value of \(x\) when it reaches 100.

15.2: Retire A Millionaire? (15 minutes)

Activity

In this activity, students use the natural logarithm to solve problems in a financial context. They first evaluate an exponential function with base \(e\) for different values of the input. After making sense of what a logarithm means in the given situation, they use the natural logarithm to solve an exponential equation by rewriting it as a logarithm and answer questions about the situation.

The financial context presented here is unrealistic and the numbers chosen are simple. The intent is to keep the function very straightforward at this stage so that students can better understand what a natural logarithm means in the given situation (MP2). Later, students will encounter more realistic contexts and numbers.

Students engage in aspects of mathematical modeling as they make some assumptions about the account and the age of retirement in the last question (MP4). If students choose to use graphing technology to solve \(1 \boldcdot e^{(0.06t)}=1000\), they are using appropriate tools strategically (MP5).

Launch

Arrange students in groups of 2. Give students a few minutes of quiet think time for the first two questions and then ask them to discuss their responses with their partner.

Pause the class for a discussion before students proceed to the last question. Make sure students understand that the expression \(\ln 5\) does not represent the time when the account will have 5 thousand dollars, because the exponent that we’re trying to find is \(0.06t\) and not just \(t\). In other words, we’re trying to solve \(e^{(0.06t)} = 5\), rather than \(e^t=5\). Once we know the value of \(0.06t\), which is \(\ln 5\), it will need to be divided by 0.06 to find \(t\).

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses for the last question. At the appropriate time, give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How did you get . . . ?”, “Why did you . . . ?”, and “Did you consider . . . ?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify why the account balance will not reach \$1,000,000 by the time they reach retirement.
Design Principle(s): Optimize output (for justification); Cultivate conversation
Representation: Develop Language and Symbols. Display or provide charts with symbols and meanings that show the relationship between the notation for a natural logarithm and the form \(f(x)=e^x\). Use familiar numbers, such as those in the warm-up. For example, displaying \(e^{4.6}=100\) and \(\ln100=4.6\) will help students reference prior knowledge during the activity. 
Supports accessibility for: Conceptual processing; Memory

Student Facing

The expression \(1 \boldcdot e^{(0.06t)}\) models the balance, in thousands of dollars, of an account \(t\) years after the account was opened. 

  1. What is the account balance:
    1. when the account is opened?
    2. after 1 year?
    3. after 2 years?
  2. Diego says that the expression \(\ln 5\) represents the time, in years, when the account will have 5 thousand dollars. Do you agree? Explain your reasoning.
  3. Suppose you opened this account at the beginning of this year. Assume that you deposit no additional money and withdraw nothing from the account. Will the account balance reach $1,000,000 and make you a millionaire by the time you reach retirement? Show your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Noah is 15 years old and wants to retire a millionaire when he is 60. If he invests \$1,000 today, what interest rate would he need to achieve this goal?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

The final question about the account balance reaching a million dollars before retirement requires some assumptions about the age of retirement. For students who struggle to get started on this question, two good approaches to encourage include:

  • making an estimate of retirement age and checking the balance of the account at that time
  • using technology to find out about when the account balance reaches one million dollars

Activity Synthesis

Invite students to share their responses and strategies for the last question. Make sure students see these approaches:

  • writing \(1 \boldcdot e^{(0.06t)}=1,\!000\) and solving using a natural log
  • using technology to graph \(y=1 \boldcdot e^{(0.06t)}\) and estimating from the graph the \(t\)-coordinate for which the \(y\)-coordinate is 1,000

If no students chose to graph to solve the question, demonstrate how to do so. Highlight the connection between the point of intersection of the graphs and the solution to \(1 \boldcdot e^{(0.06t)}=1,\!000\) calculated using a natural log.

Conclude the discussion by noting that if it were possible to retire a millionaire by depositing 1,000 dollars and leaving it untouched until retirement, it is likely that there would be many more retired millionaires. If time allows, ask students to determine what interest rate is necessary to turn \$1,000 into \$1,000,000 over 50 years. (Almost 14%.)

15.3: Cicada Population (15 minutes)

Activity

This activity gives students a chance to solve exponential equations in context by using a logarithm and by graphing.

Students have previously used graphs to estimate solutions to exponential equations. To find the input of a function that produces a certain output, they have primarily relied on visual inspection of the point when the graph reaches that value. Here they see that the estimation can be made more explicit and precise by graphing a horizontal line with a particular value, locating the intersection of the exponential function and that line, and then finding the coordinates of that intersection (by estimating or by using technology). This work previews further exploration on using the intersection of graphs to solve exponential problems in upcoming lessons.

Launch

Reading: MLR6 Three Reads. Use this routine to support reading comprehension, without solving, for students. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers. (The population of cicadas is growing at an exponential rate.) Use the second read to identify quantities and relationships. Ask students what can be counted or measured without focusing on the values. Listen for and amplify the important quantities that vary in relation to each other in this situation: population of cicadas and the number of weeks since the population was first measured. After the third read, ask students to brainstorm possible strategies to answer the first question. 
Design Principle: Support sense-making
Action and Expression: Internalize Executive Functions. To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Look for students who are rearranging the equation into the natural logarithm form to solve for \(w\), and check to make sure students are correctly inputting the equation into their graphing technology.
Supports accessibility for: Organization; Attention

Student Facing

cicada on a stem

A population of cicadas is modeled by a function defined by \(f(w) = 250 \boldcdot e^{(0.5w)}\) where \(w\) is the number of weeks since the population was first measured.

  1. Explain why solving the equation \(500 = 250 \boldcdot e^{(0.5w)}\) gives the number of weeks it takes for the cicada population to double.
  2. How many weeks does it take the cicada population to double? Show your reasoning.
  3. Use graphing technology to graph \(y=f(w)\) and \(y = 100,\!000\) on the same axes. Explain why we can use the intersection of the two graphs to estimate when the cicada population will reach 100,000.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may need help understanding the equation for \(f\). Ask questions like, “What does the 250 of \(f(w) = 250 \boldcdot e^{(0.5w)}\) tell us about the initial population?” (It was first measured to be 250.) and “How is the population changing? By what factor?” (It is growing by a factor of \(e^{0.5}\) or about 1.65 every week.) to help them get started.

While students are not asked to find the intersection of the two graphs in the final question, it is important to see this intersection point in order to appreciate the value of the graph for determining when the population reaches a certain level. Encourage students to choose a different domain or range if they cannot see where the two graphs meet, such as by adjusting the graphing window to the following boundaries: \(0\lt w \lt 16\) and \(0\lt y\lt 150,\!000\).

Activity Synthesis

Focus the discussion on the last question. Make sure students see why the intersection of the two graphs tells us the \(w\)-value when \(y\) is 100,000. We can use graphing technology to identify the \(w\)-coordinate of that intersection and obtain a more precise estimate than if we had just visually inspected the \(w\)-value when \(f(w) = 100,\!000\).

Lesson Synthesis

Lesson Synthesis

This lesson integrates the work with exponential functions and the work finding unknown exponents using logarithms. To help students make connections between these themes, here are some possible questions for discussion:

  • “How are the functions and problems in this lesson like those we’ve seen in the past?” (They involve contexts characterized by exponential growth and can be represented by expressions in the form of \(a \boldcdot b^x\). We can use graphs to help us reason about the situations.)
  • “How are they different?” (They are written using base \(e\) and involve finding missing exponents. The questions are about finding the input of the functions or finding unknown exponents. The questions can be answered by solving equations and using logarithms, or by graphing two equations and finding their intersection.)
  • “How do we solve an exponential equation by graphing?” (We graph two equations—one to represent the equation to be solved and another to represent the \(y\)-value that is the solution—and then find their intersection.)
  • “What are some benefits of finding unknown exponents by solving equations? What are some benefits of finding unknown inputs by finding the intersection of two graphs?” (Equations can give an exact answer, written in log notation, without having to graph. Graphs can be efficient if technology is available, as it skips the steps needed to solve an equation. Graphing may also help us visualize what’s happening with the quantities and see if the solution is reasonable.)

15.4: Cool-down - A Moldy Surface (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

We can use the natural logarithm to solve exponential equations that are expressed with the base \(e\).

Suppose a bacteria population is modeled by the equation \(f(h) = 1,\!000 \boldcdot e^{(0.5h)}\), where \(h\) is the number of hours since the population was first measured. When will the population reach 500,000?

One way to answer this is to solve the equation \(1,\!000 \boldcdot e^{(0.5h)} = 500,\!000\), which is when \(e^{(0.5h)} = 500\).

The natural logarithm tells us the exponent to which we raise \(e\) to get a given number, so \(0.5h =\ln 500\). This means \(h = \frac{\ln 500}{0.5}\) or about 12.4, so it takes 12.4 hours (or 12 hours and 24 minutes) for the population to reach 500,000.

We can also use a graph to solve an exponential equation. To solve \(1,\!000 \boldcdot e^{(0.5h)} = 500,\!000\), we can graph \(y = 1,\!000 \boldcdot e^{(0.5h)}\) and \(y=500,\!000\) on the same coordinate plane and find the point of intersection.

Coordinate plane, time since measurement in hours, population in thousands. Exponential function through 0 comma 1 thousand, and 12 point 4 comma 500 thousand. Dotted line at y = 500 thousand.

The graph shows us that the bacteria population reaches 500,000 when the input value is a little over 12, or about 12 hours after the population was first measured.