# Lesson 10

Interpreting and Writing Logarithmic Equations

## 10.1: Reading Logs (5 minutes)

### Warm-up

In this warm-up, students work in groups of 2 and take turns reading and interpreting logarithmic equations. Although the logarithms students have seen so far are in base 10, here they encounter simple logarithms in base 2 and base 5, which prompt them to transfer their understanding about the structure of base-10 logarithms to these other bases.

### Launch

Arrange students in groups of 2. Ask them to take turns reading and interpreting the logarithmic expressions in the activity to each other.

### Student Facing

The expression \(\log_{10} 1,\!000 = 3\) can be read as: “The log, base 10, of 1,000 is 3.”

It can be interpreted as: “The exponent to which we raise a base 10 to get 1,000 is 3.”

Take turns with a partner reading each equation out loud. Then, interpret what they mean.

- \(\log_{10} 100,\!000,\!000 = 8\)
- \(\log_{10} 1 = 0\)
- \(\log_2 16 = 4\)
- \(\log_5 25 = 2\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Invite students to share how they thought about reading each equation, focusing on the equations in base 2 and base 5. Some students may notice that those log equations share the same structure as those in base 10 and make use of that structure to interpret the equations. Highlight their observations.

## 10.2: Base 2 Logarithms (15 minutes)

### Activity

This activity extends students’ understanding of logarithms to include logarithms in another base. Students analyze patterns in a base 2 logarithm table and notice that it can be interpreted the same way as the base 10 table, except that this time the values in the right column are the exponents in expressions with base 2 (MP7).

Students use the table to evaluate base 2 log expressions and to solve simple exponential equations in base 2. They continue to reason precisely about the meaning of each parameter in the equations (MP6).

The work here prepares students to see an equation in the form of \(b^y=x\) and one in the form of \(\log_b x = y\) as equivalent, an understanding that they will develop in the next activity.

As students discuss their thinking with a partner, monitor for those who can clearly articulate the meanings of given log expressions and equations to share during the whole-class discussion.

### Launch

Arrange students in groups of 2. Display the base 2 logarithm table for all to see. Ask students to discuss with a partner why it makes sense that \(\log_2 8\) has a value of 3 and \(\log_2 1\) has a value of 0. Before students start working on the task, ensure they can articulate that \(\log_2 8\) has a value of 3 because \(2^3 = 8\) and \(\log_2 1\) has a value of 0 because \(2^0 = 1\).

Give students quiet work time and then time to share their work with a partner.

*Speaking, Reading: MLR5 Co-Craft Questions.*Use this routine to help students analyze patterns and interpret a base 2 logarithm table. Display the four tables, without the accompanying questions. Ask students to write down possible mathematical questions that could be asked about the tables. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving solving a logarithmic expression using the table or those curious about the connection between logarithmic and exponential equations. This will help students create the language of mathematical questions before feeling pressure to produce solutions.

*Design Principle(s): Maximize meta-awareness; Support sense-making*

*Representation: Internalize Comprehension.*Use color coding and annotations to highlight connections between representations in a problem. For example, demonstrate annotating the table by translating the expressions from the table into the equivalent exponential equation. While annotating, demonstrate self-verbalization of making connections with the table. Read the expression, and translate between both representations out loud. For example, highlight the row in which \(x=16\) and note, “When \(x\) is 16, the output of \(\log_{2}(x)\) is 4.” Then, while annotating \(2^4=16\), “That means that if we raise base 2 to the 4th power, the result is 16.” Encourage students to replicate the strategy as they continue to work.

*Supports accessibility for: Visual-spatial processing*

### Student Facing

\(x\) | \(\log_2 (x)\) |
---|---|

1 | 0 |

2 | 1 |

3 | 1.5850 |

4 | 2 |

5 | 2.3219 |

6 | 2.5850 |

7 | 2.8074 |

8 | 3 |

9 | 3.1699 |

10 | 3.3219 |

\(x\) | \(\log_2 (x)\) |
---|---|

11 | 3.4594 |

12 | 3.5845 |

13 | 3.7004 |

14 | 3.8074 |

15 | 3.9069 |

16 | 4 |

17 | 4.0875 |

18 | 4.1699 |

19 | 4.2479 |

20 | 4.3219 |

\(x\) | \(\log_2 (x)\) |
---|---|

21 | 4.3923 |

22 | 4.4594 |

23 | 4.5236 |

24 | 4.5850 |

25 | 4.6439 |

26 | 4.7004 |

27 | 4.7549 |

28 | 4.8074 |

29 | 4.8580 |

30 | 4.9069 |

\(x\) | \(\log_2 (x)\) |
---|---|

31 | 4.9542 |

32 | 5 |

33 | 5.0444 |

34 | 5.0875 |

35 | 5.1293 |

36 | 5.1699 |

37 | 5.2095 |

38 | 5.2479 |

39 | 5.2854 |

40 | 5.3219 |

- Use the table to find the exact or approximate value of each expression. Then, explain to a partner what each expression and its approximated value means.
- \(\log_2 2\)
- \(\log_2 32\)
- \(\log_2 15\)
- \(\log_2 40\)

- Solve each equation. Write the solution as a logarithmic expression.
- \(2^y = 5\)
- \(2^y=70\)
- \(2^y=999\)

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

As in the previous, students may be overwhelmed with the tables. Point out that the second column has the values \(\log_{2}(x)\) and so students need to find the appropriate value of \(x\) in the first column. For students not sure where to start when asked to solve the equations, ask them to return to the work of the warm-up and consider the meaning of \(\log_2(x)\) (the power \(2\) needs to be raised to in order to give a value of \(x\)).

### Activity Synthesis

Select previously identified students to share their responses and reasoning. Focus the discussion on two key ideas:

- In the equation \(2^y = 8\), we know that the value of \(y\) is 3 because \(2^3=8\), but we also know that the value of \(y\) is \(\log_2 8\), because by definition, \(\log_2 8\) is the exponent to which 2 should be raised to get 8. So we can express the solution as either 3, or \(\log_2 8\), as the two are equal.
- When solving exponential equations, the solutions expressed in logarithms are exact solutions, whereas those written in decimals are approximations. For example, the solution to \(2^y = 8\) is \(\log_2 8\) or 3. In this case, both are exact solutions. For \(2^y = 5\), the solution \(\log_2 5\) is exact, but the value we see in the base 2 log table, 2.32193, is not. If we raise 2 to the exponent of 2.32193, the result is not exactly 5.

## 10.3: Exponential and Logarithmic Forms (15 minutes)

### Activity

In this activity, students make explicit connections between equivalent equations in exponential form and in logarithmic form. They also write equations to represent descriptions of exponential relationships. Working across different forms and representations reinforces students’ understanding of the meaning of logarithms. Along the way, they continue to attend carefully to the meaning of each parameter in the equations they write (MP6).

After converting a series of numerical equations from one form to the other, students generalize the equivalence of the two forms algebraically (MP8). Note that there are restrictions on the parameter \(b\): the base cannot be negative, and we don’t typically look at bases that are less than 1. Students don’t need to concern themselves with this at the moment, however.

The activity includes equations in various bases, but students should not be assessed on their readiness to work in bases other than 2 and 10.

### Launch

Display the following two equations for all to see: \(\displaystyle 2^3 = 8\) \(\displaystyle \log_2 8 =3\)Tell students that the former is written in *exponential form*, the latter is in *logarithmic form*, and the two equations are equivalent.

Explain that the two equations represent the same relationship between a base, an exponent, and the value of that base after it is raised to the exponent. Consider articulating the meaning of each equation verbally:

- \(2^3 = 8\) can be interpreted as: “Raising 2 to exponent 3 gives 8.”
- \(\log_2 8 =3\) can be interpreted as: “The exponent to which we raise a base 2 to get 8 is 3.”

*Representation: Internalize Comprehension.*Invite students to use self-verbalization strategies to internalize comprehension. Encourage students to quietly, but audibly, repeat the phrasing for the expressions as they work through the table. Display sentence frames for each form to support students as they work. For example, “I notice that . . .” (exponential form) and “If _____ then _____ because . . . .” (logarithmic form). Encourage students to use the phrasing demonstrated in the launch: “Raising 2 to exponent 1 gives 2,” and “The exponent to which we raise a base 3 to get 81 is 4.” Invite students to use color coding on their tables to highlight connections between representations as they work.

*Supports accessibility for: Visual-spatial processing*

### Student Facing

These equations express the same relationship between 2, 16, and 4:

\(\displaystyle \log_2 16 = 4\)

\(\displaystyle 2^4 = 16\)

- Each row shows two equations that express the same relationship. Complete the table.
exponential form logarithmic form a. \(2^1= 2\) b. \(10^0=1\) c. \(\log_3 81 = 4\) d. \(\log_5 1 =0\) e. \(10^\text{-1}=\frac{1}{10}\) f. \(9^\frac12=3\) g. \(\log_2 \frac18= \text - 3\) h. \(2^y=15\) i. \(\log_5 40 = y\) j. \(b^y = x\) - Write two equations—one in exponential form and one in logarithmic form—to represent each question. Use “?” for the unknown value.
- “To what exponent do we raise the number 4 to get 64?”
- “What is the log, base 2, of 128?”

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

- Is \(\log_{2}(10)\) greater than 3 or less than 3? Is \(\log_{10}(2)\) greater than or less than 1? Explain your reasoning.
- How are these two quantities related?

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

For students who struggle writing the exponential and logarithmic equations, focus their attention on the two given equations \(2^4 = 16\) and \(\log_2(16) = 4\). How are they related? Highlight that the logarithm gives the solution to the exponential equation \(2^? = 16\). Conversely, given an exponential equation like \(2^4 = 16\) this tells us that \(\log_2(16) = 4\) since it gives the exponent 2 needs to be raised to in order to give 16.

### Activity Synthesis

Make sure students understand the connections between the two forms. Consider annotating the parameters in the two forms of equations to help illustrate the connections:

Explain to students that base 10 logarithms are often written without the subscript 10. For example, \(\log_{10} 1,\!000=3\) may be written as \(\log 1,\!000 = 3\), which is equivalent to \(10^3=1,\!000\). When no base is specified in a logarithm, it is understood that the base is 10. In all other logarithms, the base is shown.

*Speaking: MLR8 Discussion Supports.*Use this routine to amplify the language students use to explain why each pair of exponential and logarithmic equations express the same relationship. After students share a response, invite them to repeat their reasoning using mathematical language relevant to the lesson such as base, exponent, and value of the base after it is raised to the exponent. For example, say, "Can you say that again, using the words ‘base’ and ‘exponent’?" Consider inviting other students to repeat these phrases to provide additional opportunities for all students to produce this language. This will help students make connections between equivalent equations in exponential and logarithmic form.

*Design Principle(s): Support sense-making*

## Lesson Synthesis

### Lesson Synthesis

To reinforce the key ideas in the lesson, discuss questions such as:

- “How are \(\log_2 60\), \(\log_5 60\), and \(\log_{10} 60\) alike? How are they different?” (They all represent the exponent of a power expression that has a value of 60 and they are all logarithms. They express the exponent for powers with different bases.)
- “Is the equation \(\log_{100} 10 = \frac12\) true? How do you know?” (Yes. The 100 is the base, the \(\frac12\) is the exponent, and 10 is what we get if we raise 100 to exponent \(\frac12\), and \(100^\frac12 = \sqrt {100} = 10\).)
- “How might we write the solution to \(2^y = 36\)?” (\(\log_2 36\) or 5.169925)
- “Does it matter if we write the solution as \(\log_2 36\) or 5.169925? Is there a difference?” (The solution written as a logarithm is exact. The other solution is an approximation.)
- “We learned that \(7^2 = 49\) and \(\log_7 49=2\) are equivalent. How would you explain to someone why we consider these two equations as equivalent, even though they don’t look alike?” (They are two ways to represent the same relationship. \(\log_7 49\) tells us the exponent by which we raise a base 7 to get 49, and that exponent is 2.)
- “Suppose some number \(n\) raised to exponent \(y\) has a value of 200. How can we express this relationship in exponential form and logarithmic form?” (\(n^y = 200\) and \(\log_n 200 = y\))

## 10.4: Cool-down - Writing in Different Forms (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

Many relationships that can be expressed with an exponent can also be expressed with a logarithm. Let’s look at this equation: \(\displaystyle 2^7 = 128\) The base is 2 and the exponent is 7, so it can be expressed as a logarithm with base 2:

\(\log_2 128 = 7\)

In general, an exponential equation and a logarithmic equation are related as shown here:

Exponents can be negative, so a logarithm can have negative values. For example \(3^{\text-4} = \frac{1}{81}\), which means that \(\log_3 \frac{1}{81} = \text-4\).

An exponential equation cannot always be solved by observation. For example, \(2^x = 19\) does not have an obvious solution. The logarithm gives us a way to represent the solution to this equation: \(x=\log_2 19\). The expression \(\log_2 19\) is approximately 4.25, but \(\log_2 19\) is an exact solution.