Transcript Introduction to Algebra

Logarithmic Scales
The Richter Scale
for
Earthquakes
The Richter Scale

For fixed intensity x 0 (as measured with
a seismometer) the ratio of the seismic
intensity x of an earthquake, relative to x 0 ,
is measured by
R = log 10 (


)
This is the famous Richter Scale use to
measure relative “strength” of earthquakes
x
For x > x0 note that x > 1
0
so
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x
x0
R = log 10 (
x
x0
Logarithmic Richter Scale
)>0
2
The Richter Scale
R = log 10 (
x
x0
)>0
Question:
How fast does R grow ?
If R increases by 1 what is the change in x ?
What change in x changes R by 1 ?
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Richter Scale Comparisons

In 1992 the Landers earthquake produced
a Richter scale value of 7.3 compared with
the 1994 Northridge earthquake which hit
6.7 on the Richter scale

How much more powerful was the Landers
earthquake, expressed as a ratio ?
That is, find L / N where
L = Landers intensity and
N = Northridge intensity
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Richter Scale Comparisons
Find L / N where
L = Landers Richter value and
N = Northridge Richter value
N
L
RL = log 10 x = 7.3 and RN = log 10 x = 6.7
0
0
N
L
7.3
By definition x = 10
and x = 10 6.7
0
0
Thus
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L = x 0 10 7.3 and N = x 0 106.7
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Richter Scale Comparisons
L = x 0 10 7.3 and N = x 0 106.7
Thus the ratio of L to N is
x0 107.3
L
7.3 – 6.7 = 100.6 ≈ 3.981
10
=
=
N
x0 106.7
Thus L = 3.981N , i.e. 3.981 times as strong
as Northridge
How much stronger was Landers than the
smallest recorded earthquake with Rs = 4.8 ?
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Richter Scale Comparisons
How much stronger was Landers than the
smallest recorded earthquake with Rs = 4.8 ?
x0 10 7.3
L
7.3 – 4.8 = 10 2.5 ≈ 316
=
=
10
S
x0 10 4.8
The Landers quake was 316 times stronger
than the smallest recorded quake
How do various well-publicized earthquakes
compare in intensity ?
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Earthquake Comparison Ratios

2010 Haiti
Richter
Scale
6.1
Relative Strength Ratios
3.98

1994 Northridge 6.7
7.94
2.00

2010 Haiti
7.0
3.98
501.2
2.00

1992 Landers
7.3
63.10
?
31.62

2010 Chile
8.8
199.53
100.0
3.16

2004 Indonesia 9.3
1.58

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1960 Chile
9.5
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Earthquake Comparison Graph
R
10.0
●Chile
9.0
8.0
7.0
6.0
5.0
●
●
●
R
Landers
7.3
Haiti
7.0
● Chile
9.5
9.3
8.8
8.0
7.0
4.0
3.0
●Indonesia
6.0
2.0
●Landers
●
Haiti
●
7.0
Northridge
●Haiti 6.7
6.1
7.3
Intensity
x 106
x Intensity
x 106
20
1.0
10
1,000
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2,000
Logarithmic Richter Scale
3,000
4,000
x
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Earthquake Comparison Ratios

2011 Hawaii
Richter
Scale Relative Strength Ratios
4.5
79.43

2011 Hawaii
6.4
316.2
3.981

2010 Haiti
7.0
31,622.8
398.1
63.10

2010 Chile
8.8
100.0
1.585

2011 Japan
?
1,258.9
9.0
1.995

2004 Indonesia 9.3
3.162
1.585

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1960 Chile
9.5
Logarithmic Richter Scale
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Earthquake Comparison Ratios

2011 Hawaii
Richter
Scale Relative Strength Ratios
6.4
3.162

1989 California
6.9
3.981
1.259

2010 Haiti
7.0
398.1
125.9
63.10

2010 Chile
8.8
1.585

2011 Japan
?
100.0
398.1
9.0
1.585

1964 Alaska
9.2
3.162
1.995

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1960 Chile
9.5
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Think about it !
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