Activity 4.9 Logarithmic Scales
Overview:
The activity asks students to plot the size of several animals on a number line and then plot the
log of their sizes and describe which is a better representation of the data and why it is better.
Estimated Time Required: The activity should take approximately 15 minutes.
Technology: Scientific Calculator
Prerequisite Concepts:
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Linear scale
Logarithmic scale
Definition of the common logarithmic function
Discussion:
The students should see that linear scales are not adequate for displaying data sets that involve a
large range of values. Explain that a linear scale requires that the tick marks represent equal
increments. Consider a few different linear scales and demonstrate how it is impossible to
display the entire data set.
Have students label the tick marks of the second number line as powers of ten: 100 ,101,102 , .
Then ask the students to plot the data on the new scale. To plot a point requires the students to
write each size as a power of ten. For example, to write 200 as a power of ten requires the
solution to the equation 10t  200 , which is t  2.30103. The size of a cat would be plotted at
approximately 2.3 on the log scale.
Remind students of the definition of the common logarithm function and relate the definition to the
plots that they have constructed. It is very important that students understand the exact meaning
of the expression “log x.” Require students to know the following definition:
log x is the exponent to which 10 must be raised to produce x.
Use some of the computations they made to illustrate the definition.
Even though all calculators have a log button, the students should understand what the output
value means and they should be able to estimate logarithm values without a calculator. For
example, log 0.45 is between –1 and 0 because 101 = 0.1 and 100 = 1.
Activity 4.9 Logarithmic Scales
Animals and their respective sizes are given in the table below.
Animal
Virus
Bacterium
Human Cell
Ant
Hummingbird
Domestic Cat
Wolf (with tail)
Thresher Shark
Giant Squid (with tentacles)
Sequoia
Size (cm)
0.0000005
0.0002
0.002
0.8
12
60
200
600
2200
7500
1. Plot the sizes for each animal on a linear scale. Number the increments on your number line
and label each point.
2. Now, find the log of each animal’s size.
Animal
Virus
Bacterium
Human Cell
Ant
Hummingbird
Domestic Cat
Wolf (with tail)
Thresher Shark
Giant Squid (with tentacles)
Sequoia
Size (cm)
0.0000005
0.0002
0.002
0.8
12
60
200
600
2200
7500
Log of Size
Plot the log of size values on a logarithmic scale. Number the increments on your number line
and label each point.
Which scale (linear or logarithmic) was better to use for the given data? Why
Complete the sentence:
When plotting data, logarithmic scales are better than linear scales when the data is
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