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Lesson 2.2
Pre-Calculus: Modeling Our World
Exponentials & Logarithms
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Logarithmic Scale
LESSON 2.2
he chapter introduction raised the issue of geologists’ classification of earthquakes.
There are many factors that go into creating an effective scale for measuring the
size of an earthquake. In the following activity, you will explore some of these factors,
and how mathematics can be applied to this measurement problem.
T
ACTIVITY 2.3
RELATIVE SIZES AND
ORDERS OF MAGNITUDE
ACTIVITY 2.4
LOGARITHMIC SCALE
EXERCISES 2.2
Activity 2.3
Table 2.16 shows the computed intensities for a portion of the
earthquake activity for Central America and California on May 1,
1997; other earth movements took place on that day, but were not
sizeable events. (Note: For the purposes of this chapter, no units
are used with intensity values; however, they are directly
proportional to energy values discussed in the Introduction.)
1. According to an ancient legend, really big earthquakes
happen at sunset. A geologist investigating the scientific
merit of that claim decides to graph seismic energy versus
time of day for the data in Table 2.16.
Time
Latitude Longitude
Intensity
Time
Latitude Longitude
Intensity
3:30 A.M. 32.2°N
115.8°W
2.143 X 1013
4:41 P.M. 18.7°N
106.9°W
2.518 X 1015
5:08 A.M. 32.0°N
115.8°W
2.244 X 1013
5:00 P.M. 33.5°N
118.0°W
2.000 X 1013
6:22 A.M. 34.4°N
118.6°W
1.233 X 1014
5:17 P.M. 34.1°N
116.4°W
2.698 X 1013
6:22 A.M. 34.4°N
118.7°W
2.000 X 1013
8:12 P.M. 34.0°N
116.8°W
2.576 X 1013
8:02 A.M. 32.2°N
115.8°W
2.891 X 1013
8:41 P.M. 39.6°N
122.0°W
1.517 X 1014
9:45 A.M. 37.5°N
121.7°W
3.991 X 1013
9:22 P.M. 35.8°N
117.6°W
2.193 X 1013
11:37 A.M. 19.0°N
107.4°W
2.518 X 1018 10:07 P.M. 36.0°N
120.6°W
1.262 X 1014
12:52 P.M. 39.4°N
119.8°W
1.262 X 1014 10:36 P.M. 38.9°N
122.8°W
2.000 X 1013
4:25 P.M. 18.7°N
107.1°W
3.991 X 1015
TABLE 2.16.
Earthquake Activity–—5/1/97 (adapted from Council of the National Seismic System data).
Lesson 2.2
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Pre-Calculus: Modeling Our World
Chapter Two
a) Describe how to scale the horizontal axis to represent the
earthquake times.
b) Describe how to scale the vertical axis to represent the seismic
energy intensity for the earthquakes.
c) Use a graphing calculator to make a scatter plot of the intensity
versus time data in the table. Describe your graph. How can you
adjust the window or data to make the graph more informative?
d) According to these data, do really big earthquakes occur only at
sunset? Explain.
The geologist Charles Richter was faced with the problem of
constructing a scale that would handle a large range of values, such as
the earthquake data for May 1, 1997. The intensity of an earthquake is
proportional to its energy output. Richter used a description called
magnitude, calculating M = log(I/I0). Here I0 is the intensity of the
smallest detectable earthquake at that time (intensity 2.0 x 1011). In
answering Items 2 and 3, use a spreadsheet or statistical lists on a
calculator to investigate how the magnitude equation handles the
problem of scaling intensities.
Charles Richter
2. First, perform only the calculation inside the parentheses (I/I0), and
record your answers in a copy of Table 2.17. This ratio, I/I0, is the
relative intensity of the quake.
a) How does this first step of the magnitude calculation affect the
values of intensity given in the table? What kind of
transformation is created?
I
2.143 x 1013 2.244 x 1013 1.233 x 1014 2 x 1013 2.891 x 1013 3.991 x 1013 2.518 x 1018 1.262 x 1014
I/I0
I
3.991 x 1015 2.518 x 1015 2 x 1013 2.698 x 1013 2.576 x 1013 1.517 x 1014 2.193 x 1013 1.266 x 1014 2 x 1013
I/I0
TABLE 2.17.
Calculation of
relative intensities.
b) What is the contextual meaning of a computed ratio of 100 in
part (a)? In general, what is the meaning of the calculation (I/I0)?
c) Graph relative intensity vs. time. Does calculating relative
intensity solve Richter’s problem of creating a useful scale for
displaying earthquake data? Explain.
Pre-Calculus: Modeling Our World
Exponentials & Logarithms
Lesson 2.2
3. Now, using a scientific calculator capable of finding logs, carry out
the second step of the magnitude calculation. Record your answers
in a table like Table 2.18.
a) What does the log step of the calculation do to the relative
intensities?
I/I0
107.2
112.2
Magn. 2.03
11M
b) Graph magnitude vs. time. Does the use of logs in the magnitude
calculation solve Richter’s problem of creating a useful scale for
displaying earthquake data? Explain.
c) Based on the pattern in your scatter plot, do you think there were
any pre- or after-shocks? If so, identify them.
4. Understanding the individual steps in the magnitude calculation
provides insight into what Richter was trying to accomplish.
Examine some of the details in greater depth to become more
familiar with log calculations.
a) An earthquake with a relative intensity of 100 (=1.00 x 102) has a
magnitude value of 2.00. Where did the 2 come from? What does
it really mean? Under what circumstances would an earthquake
magnitude be between 2 and 3? That is, when would it be 2 and a
decimal, like 2.xx?
b) Use trial and error to find relative intensities that produce Richter
numbers of exactly 3, 4, 5, ... , 10. Describe any pattern you see.
c) May 1, 1997 was chosen because an earthquake measuring 7.1 on
the Richter scale took place. Look back over the calculations, and
explain where the 7 came from.
d) What is the meaning of the integer part of a magnitude number?
5. Draw a number line and number its scale from 0 to 10 by ones.
Label the line Richter number.
a) Indicate on the number line where each earthquake from May 1,
1997 belongs, according to its Richter number.
TABLE 2.18.
Calculating magnitude
from relative intensity.
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Lesson 2.2
Pre-Calculus: Modeling Our World
Chapter Two
b) Re-scale the number line (write the new scale markings below
your Richter number scale markings) so that it indicates the
relative intensities as well as the Richter numbers. Describe the
relationship between the two scales.
RELATIVE SIZES AND ORDERS OF MAGNITUDE
The calculation of relative intensity in Activity 2.3 illustrates a pretty
standard technique in mathematics for moving from the question “how
big is it” to “how much bigger is it than . . ?”. Establishing relative size
requires some kind of standard for comparison, and Richter used the
smallest possible earthquake detectable at that time. This eliminated
decimal values smaller than one. In addition, using ratios allowed him to
compare two measurements directly, not just to compare to a standard.
For example, if one earthquake had a relative intensity of 100,000 and
another 1000, the first would be 100 times as large as the second, since
100,000 ÷ 1000 = 100.
The data reported by the Council of the National Seismic System
included only earthquakes of magnitude 2.0 or larger. There were many
more earthquakes with less energy that were detected but not reported.
Even then, there were 17 seismic events in that region of the continent in
one day. This represents a typical day, except for the “big one” (M = 7.1).
Modern apparatus is sensitive enough to detect earth movements as
small as a rock breaking on a lab table (M = –1.5).
The Richter scale for describing earthquake intensities is an example of a
magnitude scale, named after the fact that the scale expresses the order
of magnitude (or power of 10) for the number, not the actual number. It
is used whenever the quantities being represented have values over a
large range of sizes. The problem of graphing the data in Activity 2.3
revealed that a uniform scale does not adequately depict numbers over
several orders of magnitude. A small scale interval shows all the values
correctly, but requires a very large grid, while a large scale interval
reduces small values to essentially zero. Instead, using a scale that
represents the various powers of ten retains relative sizes, and places all
numbers on a more useful scale. The mathematical tool for changing the
scale in such a way is the logarithm function, or “log” for short. The
logarithm converts any positive number into its “power-of-ten
exponent”. That’s why the Richter scale is more completely referred to
as the Richter magnitude scale.
Exponentials & Logarithms
Pre-Calculus: Modeling Our World
EXAMPLE 1
Without using your calculator, evaluate the following logarithms:
a) log(100)
b) log(1/10)
c) log(50)
SOLUTION:
Since 102 = 100, log(100) = 2. Likewise, 10–1 = 1/10, so log(1/10) = –1.
10 < 50 < 100, so log(50) is between 1 and 2. The problem of obtaining a
better approximation will be revisited.
The key to constructing a magnitude scale is to use equally-spaced scale
markings to represent increasingly large intervals. For example, the
Richter magnitude interval [0, 1] represents relative intensities from
1–10, while the Richter magnitude interval [1, 2] represents relative
intensities from 10–100. A magnitude scale accomplishes two things.
First, the resulting numbers are conveniently small and convey
information about the order of magnitude. Second, relative intensity
calculations reduce to subtraction.
EXAMPLE 2
Consider two earthquakes with Richter numbers 7 and 4. Determine
how much more powerful the stronger is than the weaker.
SOLUTION.
The corresponding relative intensities are 1 x 107 and 1 x 104, so the
first is 103 times as powerful as the second. Note that 7 – 4 = 3, so the
calculation of relative strength reduces to the subtraction of Richter
numbers.
Sound is another familiar physical phenomenon that is measured (and
described) in a manner similar to that of earthquakes. Sounds are
produced when vibrating objects cause pressure waves in some medium,
Lesson 2.2
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Lesson 2.2
Pre-Calculus: Modeling Our World
Chapter Two
like air or water, that then carries the energy of the vibrations. These
pressure variations are detected as sound when a receiver, such as a
human eardrum, is caused to vibrate in a similar fashion. The loudest
sounds the human ear can detect are more than a trillion (1012) times
louder than the softest sounds. A sound’s intensity is measured as
power per unit area. The sound level (loudness) of a sound wave is
commonly specified with a value B, measured in decibels (abbreviated
dB), which is defined as:
B = 10 ⋅ log(I/I0).
I0 is the threshold for hearing, defined to be I0 = 1 x 10–12 watts/m2.
Calculations of sound levels and relative intensities are very similar to
those of earthquakes.
EXAMPLE 3
How many times more powerful is the sound of a jet plane taking off
(B = 150 dB) than the noise generated by a car traveling at 60 mph
(sound intensity at that speed is 10–4 watts/m2).
SOLUTION:
Let I2 represent the intensity of the plane, and I1 the intensity of the car.
Step 1: Find the relative intensity for the sound level of the car:
I1/I0 = 108. (Thus the sound level is (10)(8) = 80 dB.)
Step 2: Find the relative intensity for the sound level of the plane:
150 = 10 ⋅ log(I2/I0), so log(I2/I0) = 15, or I2/I0 = 1015.
Step 3: Compare the relative intensities of the plane and car:
(I2/I0)/(I1/I0) = 107. Therefore, I2/I1 = 107, and the jet engine is
10 million times louder.
Note: Subtracting the two decibel levels to compare relative intensities
again helps with the comparison, but you must also divide by 10:
150 – 80 = 70, and 70/10 = 7, so I2/I1 = 107, as before.
Other contexts in which magnitude scales are used include light
intensity and nuclear radiation.