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
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.
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.
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"
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.