# Lesson 11

Evaluating Logarithmic Expressions

## 11.1: Math Talk: Finding Values (5 minutes)

### Warm-up

This Math Talk encourages students to think about the relationship between base 10 logarithms and exponential expressions. Students have a chance to rewrite decimals, fractions, and base 10 numbers as powers of 10 in order to see which power of 10 the number is equal to and so find its base 10 logarithm. They will use this skill later in the lesson and in future lessons in order to calculate or estimate different logarithms.

### Launch

Remind students that logarithms with no specified base are base-10 logarithms.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

### Student Facing

Evaluate mentally.

1. $$\log 10$$
2. $$\log 10,\!000$$
3. $$\log 0.1$$
4. $$\log \frac{1}{1,000}$$

### Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate $$\underline{\hspace{.5in}}$$’s reasoning in a different way?”

• “Did anyone have the same strategy but would explain it differently?”

• “Did anyone solve the problem in a different way?”

• “Does anyone want to add on to $$\underline{\hspace{.5in}}$$’s strategy?”

• “Do you agree or disagree? Why?”

If not already mentioned, make sure students recall that numbers such as 0.1 and $$\frac{1}{1,000}$$ are still powers of 10 but they have negative exponents (that is, 0.1 is $$10^{\text -1}$$, and $$\frac {1}{1,000}$$ is $$\frac{1}{10^3}$$ or $$10^{\text-3}$$). This suggests that the logarithms of numbers that are less than 1 would be negative.

Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because…” or "I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

## 11.2: Log War! (20 minutes)

### Activity

In this activity, students compare logarithmic expressions as they play a game of “war” in groups of 2. Students receive cards containing logarithms in bases 2, 5, and 10. The numbers in the expressions are written in all forms, including decimals, fractions, whole numbers, and exponential expressions. To compare the logarithms on the cards, they may:

• determine if an expression is negative, zero, or positive, without finding the exact or estimated value
• estimate the value of an expression such as $$\log_2 10$$ by using logarithms with integer values
• convert decimals to fractions to help find the logarithms

As students preview the deck by putting the cards in order from least to greatest, listen for how they decide on relative values of the cards. Students should not use calculators at this point.

### Launch

Arrange students in groups of 2. The first 20 cards in the blackline master show log expressions with integer values. That last 10, which are labeled “challenge,” have non-integer values. Give each group only the first 20, the entire set of 30, or differentiate by giving some groups only the first 20 and some groups all of the cards.

Before students start playing, ask them to turn all the cards face up, and work with their partner to put them in order from least to greatest. The game will go more smoothly if students have a chance to gain familiarity with the contents of the card deck and evaluate each card in a lower-stakes interaction. This step may be skipped if students are comfortable determining the value of logarithmic expressions.

Speaking: MLR8 Discussion Supports. To help students discuss the order of the cards with their partner, provide sentence frames such as: “I know _____ because . . .”, “Why did you . . . ?”, and “How do you know . . . ?”
Design Principle(s): Optimize output (for justification); Cultivate conversation
Engagement: Provide Access by Recruiting Interest. Begin with a small-group or whole-class demonstration of how to play the game. Check for understanding by inviting students to rephrase directions in their own words. To scaffold the activity to allow for self-correction and multiple attempts, invite students to discard any cards that they may hesitate on into a separate, designated pile. Build in a few minutes for students to look through these cards after their first attempt. Then ask students to reshuffle and make a second attempt, noticing what strategies helped them to more quickly work through the more challenging cards.
Supports accessibility for: Memory; Conceptual processing

### Student Facing

Have you played the game of war with a deck of playing cards?

Your teacher will give you and your partner a set of special cards.

• Shuffle and deal the cards evenly.
• Each player turns one card face up. The card with the greater value wins the round and its player captures both cards and sets them aside.
• If you and your partner disagree about the value of a card, discuss until you reach an agreement.
• If there is a tie, each player turns another card face up. The player whose card has the greater value captures all cards (including the cards that tie).
• Play until all the cards are turned up. The player with the most cards wins.

Let the logarithm war begin!

### Student Facing

#### Are you ready for more?

Mai uses the fact that $$\sqrt{2}$$ is close to 1.25 and that $$2^3=8$$  to make the estimate $$\log_2(10) \approx 3\frac{1}{3}$$. Explain Mai’s estimate. How exact is the estimate?

### Anticipated Misconceptions

Students may struggle making estimates to compare expressions. Ask them questions such as "Is the expression greater than or less than 0? How do you know? Is the expression greater than or less than 1? How do you know?" Make sure they understand that they only need to decide which is bigger; they do not need to calculate exact values.

### Activity Synthesis

Much of the discussions will happen in groups, as students put the cards in order before playing the game and while playing the game. Invite groups to share any insightful or useful strategies for comparing any two cards that came up before or during the game.

## 11.3: Finding Logarithms with a Calculator (10 minutes)

### Activity

This activity has two aims: to familiarize students with the log function on a calculator, and to reinforce the fact that most values obtained on a calculator, just like those in a log table, are approximations.

The questions involve finding only base 10 logarithms. This is partly to focus the exploration, and partly because many calculators can evaluate only logarithms in base 10 and base $$e$$, or require the user to perform steps to change the base before calculating logarithms in other bases. Some calculators allow easily calculating logs with any base. If the devices used in your class allow for this, show students how it works.

The purpose of the last question is to prompt students to think about reasonable values of the logarithms and to not simply enter numbers into the calculator. Monitor for students with clear estimation strategies to share during the whole-class discussion. Beyond reasoning that a log value is between two whole numbers, students don't yet have a good means of making an accurate estimate, which is fine. For example, they might be able to say that $$\log 90$$ should be quite close to 2, but it is not expected that they could say how close.

### Launch

Keep students in groups of 2.

Explain to students that before calculators were widely available, finding logarithms involved using tables like the ones we saw in earlier lessons. Today, all scientific calculators have the capacity to calculate base 10 logarithms at a minimum. Some calculators can calculate logarithms in any base, making log tables essentially obsolete.

Action and Expression: Internalize Executive Functions. Chunk this task into manageable parts for students who benefit from support with organizational skills in problem solving. Check in with students after the first 2–3 minutes of work time. Invite 1–2 students to share their thinking for evaluating Tyler and Priya’s responses, emphasizing that there are multiple considerations when using technology or approximating. Record their thinking on a display and keep the work visible as students continue to work.
Supports accessibility for: Organization; Attention

### Student Facing

1. To solve the equation $$10^m= 19$$, Tyler writes the equation in the logarithmic form: $$m = \log_{10} 19$$. He then presses the “log” button on the calculator, enters the number “19,” and writes down an approximation of 1.279. Priya follows the same steps on her calculator and writes down 1.27875.
1. Experiment with your calculator until you understand how to evaluate $$\log_{10} 19$$. What value do you see on the calculator?
2. Discuss with a partner: Why might $$\log_{10} 19$$ be expressed in different ways?
2. Express the solution to each equation using a logarithm. Next, find the approximate value of the solution using a calculator.
1. $$10^m=24$$
2. $$10^n=750$$
3. Estimate the value of each expression. Explain to a partner how you made your estimate. Next, check your estimate with a calculator.
1. $$\log 90$$
2. $$\log 1,\!005$$
3. $$\log 9$$

### Activity Synthesis

Make sure students understand how the log function on their calculator works and can find log values correctly.

Invite previously selected students to share how they estimated log values before using a calculator. Emphasize that having a sense of what value would be reasonable before entering a logarithm in a calculator can help us notice any errors.

Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them respond: “I agree because . . .” or “I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. For example, a statement such as, “It’s about 2” can be restated as a question such as, “Is the log, base 10, of 90 approximately 2?” This will help students clarify their understanding of how to use the log function on a calculator.
Design Principle(s): Support sense-making

## Lesson Synthesis

### Lesson Synthesis

Encourage students to reflect on their strategies for finding or estimating logarithms. Discuss questions such as:

• “When playing the game, how did you know if the value of the expression is positive, negative, or zero?” (If the number for which we’re finding the logarithm, say $$x$$, is greater than the base, then the log value is positive. If $$x$$ is the same as the base, then the log value is 1. If $$x$$ is less than the base but greater than 1, then the log value is between 0 and 1. If $$x$$ is less than 1, then the log value is negative.)
• “When the solution to an exponential equation is estimated with a calculator or a log table and not written in log notation, how would you know if it’s reasonable? For example, is 2.675 a reasonable solution to $$2^y=18$$ or $$\log_2 18$$?” (We can mentally substitute the solution into the exponential form to check. We know that $$2^4$$ is 16, or $$\log_2 16 = 4$$, so the value of $$y$$ must be greater than 4 and cannot be around 2.675.)

## Student Lesson Summary

### Student Facing

Sometimes it’s possible to find an exact value for a logarithm. For example $$\log_2 0.125 = \text{-}3$$ because $$0.125 = \frac18$$ and $$2^{\text{-}3} = \frac18$$. Similarly, $$\log_5 625 = 4$$ because $$5^4 = 625$$.

Often times it is not possible to find an exact value, but using number sense allows us to get a reasonable estimate. Let’s say we want to find $$x$$ that makes $$10^x = 980$$ true.

We can first express $$x$$ as $$\log_{10} 980$$. Because 980 is between $$10^2$$ and $$10^3$$$$,$$ $$\log_{10} 980$$ is between 2 and 3.

But where does $$\log_{10} 980$$ lie between 2 and 3? Because 980 is much closer to 1,000 than it is to 100, $$\log_{10} 980$$ is likely a lot closer to 3 than it is to 2. This means 2.9 would be a better estimate than 2.1 would be.

We can use a calculator to verify our estimate and find that $$\log_{10} 980$$ is very close to 3, about 2.99.