## Logarithms Demystified

• Lesson
9-12
1

Before there were electronic calculators, there were logarithm tables and slide rules. In this lesson, students make and use slide rules to discover the properties of logarithms. The technique, analogous to number-line addition, reinforces the hierarchy among the operations of addition, multiplication, and exponentiation.

Prior to this lesson, students should be comfortable with the concept of a logarithm and with the properties of exponents.

Use transparent rulers from the Slide Rule overhead to show students a non-electronic calculator for addition. Line up rulers as shown in the diagram. Slide the endpoint of the top ruler to the first addend. Read the sum below the second addend. Because the top ruler is shifted 2 units to the right of the bottom ruler, the diagram below shows 2 + 4 = 6, 2 + 11 = 13, and many other addition examples in which 2 is added to another number.

To subtract 9 from 11 using the two rulers above, place the 9 on the top ruler over the 11 on the bottom ruler. Read the difference below the left end of the top ruler. Allow students to work with a partner to practice. Each student should write two simple arithmetic problems, one addition and one subtraction, and have the partner solve them using two rulers. Although these problems are trivial, the technique is the model for the unfamiliar logarithm problems that students will see later in the lesson.

Logarithm Ruler Creation

Hand out the Logarithms Demystified activity sheet. If necessary, explain to students how to find logarithms using their calculators, and then have them complete Questions 1‑3 on the activity sheet. Ask students to describe the patterns they see as they look across the tables. Note that, moving from left to right, the numbers get closer together. Point out that numbers get farther apart when a base is raised to increasing powers. Ask students to identify patterns going down the columns. They should recognize that when a number is multiplied by 10, the logarithm increases by 1.

 Logarithms Demystified Activity Sheet

Distribute the Slide Rule Template, which includes an unmarked top and bottom ruler, to each student. Have students use the data from Question 1 on the activity sheet to mark the top and bottom. They will measure the distance indicated by log n and label it with the value of n. For example, the log of 1 is 0, so the left edge — which is 0 units from the left edge — should be labeled as 1. Similarly, the log of 2 is 0.301, so the numeral 2 should be marked is at a distance of 0.301 units from the left. And so forth.

 Slide Rule Template

When all labels have been added to the unmarked rulers, the result will look similar to the image below.

Using the Slide Rule Note that the last set of rulers included on the Logarithms Demystified template shows a slide rule with appropriate markings for the logarithmic scale. You may wish to show this to students, and some students will want to transfer their markings from the Slide Rule Template to an unmarked sheet of paper, so that they can have a "clean" copy, too.

Demonstrate how to use the slide rule to add logarithms. Be sure to emphasize that the technique is the same, but the values marked on the log ruler represent logarithms, not just numbers. In other words, use these rulers to demonstrate that log 2 + log 3 = log 6. The image below shows how the rulers can be usd to show that log 5 + log 8 = log 40, log 5 + log 6 = log 30, log 5 + log 4 = log 20, and any other log addition problem involving log 5.

Do not explain the properties of logarithms at this point; just demonstrate the addition. Have students work on Questions 4 and 5 on the activity sheet. Check to see that students have the correct answers and ask students to describe any patterns they see. [The argument of the sum is the product of the arguments. Students will probably not use the word argument, but they should at least see it as the input to the log function.] Ask students to generalize this rule. Guide them to look at the examples and substitute variables for the numbers, concluding that log ab = log a + log b. Ask students to use their calculators to verify other examples, such as log 12 + log 11 = log 132.

Demonstrate that students can also subtract logarithms using the slide rule. Have them complete Questions 6 and 7 on the activity sheet. If students are having difficulty, use linear rulers until they understand the technique. As they finish, ask students to describe patterns they see in the results. They should see that the argument of the difference is the quotient of the arguments. Again, ask students to generalize the rule to be log a/b = log a – log b. Have students verify more examples with their calculators. Students who have already studied the change of base property often confuse this with the logarithm property of division. Be sure your students understand that log a/b = log a – log b is not the same as logb a = (log a)/(log b).

The power property is illustrated by repeated addition problems in Questions 8 and 9. As students realize that log 3 + log 3 + log 3 = 3 × log 3 = log 27, they may need guidance to recognize that 27 = 33.

Once students have discovered all three properties, review by listing the properties on the board. Student should generalize and record these properties using algebraic notation in Question 10.

Assessments

1. Have students create a study card or other graphic that lists and gives examples for the parallel properties of exponents and logarithms.
2. Ask students to write a paragraph explaining the use of the slide rule to demonstrate the multiplication property of logarithms.
3. Ask students to use the properties of exponents and the definition of inverse functions to derive the properties of logarithms.
4. How would you use your slide rule or the data table on your worksheet to find the log 75, log 360, or log ½?

Extensions

1. Students use the internet to research the history and uses of slide rules. They can also research other pre-electronic calculating devices such as abacuses.
2. Zipf's Law shows that the frequency of words in a document is proportional to their rank. Plotting frequency vs. rank on log paper will produce a straight line. Similar results have been found for ranking the populations of US cities.

Questions for Students

[Students will articulate these differently. Multiplication is repeated addition, division is repeated subtraction, exponentiation is repeated multiplication.]

2. Compare the laws of logarithms to the laws of exponents.

[Multiplication and addition go together: xa &mdots; xb = x(a+b) and log ab = log a + log b. Similar for division and subtraction. Discuss the idea that these properties are related because exponential and logarithmic functions are inverses of each other.]

3. Before electronic calculators, slide rules based on logarithms were used for a variety of computations including multiplication of large numbers. Why do you think they are useful for this purpose?

[The logarithms are much smaller than the actual numbers and are easier to manipulate. Discuss how the properties are useful for manipulating equations of natural phenomena like pH (acid), Richter (earthquake), and sound (decibels) because they are logarithmic.]

Teacher Reflection

• Did the slide rule help students to understand the concept of logarithm? Were students able to articulate the properties?
• Was students' prior knowledge of the properties of exponents and the definition of logarithms adequate to derive the logarithm properties?
• Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were these adjustments effective?

### Learning Objectives

Students will:
• Use rulers to add and subtract visually
• Compare logarithms for different orders of magnitude
• Create log scale rulers
• Use a simplified slide rule to discover the properties of logarithms

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.