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Mrs. Hayden
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10 – 4 Applications of Common Logs
Relative Intensity of Sound
The intensity of the faintest sound audible to the human ear is called the
threshold of hearing.
It is 10-12 watts per meters squared.
Sound intensities, I, are compared with the threshold of hearing, I o, producing a
ratio.
The relative intensity, R, of a sound in decibels (dB) is given by the function
I
I = sound intensity, I 0  the threshold
R  10 log , where
I0
of hearing
I
R  10 log 0  10 log 1  0
threshold of hearing = 0 decibels
I0
1. The intensity of a whisper is about 300 times as loud as the threshold of
hearing, I 0 . Find the relative intensity, R, in decibels.
2. The relative intensity of a vacuum cleaner is about 70 decibels. Compare this
to the threshold of hearing.
3. The relative intensity of a loud siren is about 1013 times as loud as the
threshold of hearing, I 0 . Find the relative intensity, R, of the loud siren, in
decibels.
4. The sound of a leaf blower is about 1010.5 times the intensity of the threshold of
hearing, I 0 . Find the relative intensity, R, of this leaf blower in decibels.
Mrs. Hayden
5. Suppose that the relative intensity, R, of a rock band is about 115 decibels.
Compare the intensity of this band with that of the threshold of hearing, I 0 .
6. A small jet engine produces a sound whose intensity is one billion times as
loud as the threshold of hearing. What is the relative intensity of the engine’s
sound in decibels?
7. Conversation is about 350,000 times the intensity of the threshold of hearing.
Find the relative intensity, R, in decibels.
pH level of a substance:
In chemistry, the pH of a substance is defined as
pH   log H  , where H  is the hydrogen ion concentration in
moles per liter.
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 
An acidic solution has a pH value in the range 0 < pH < 7.
An alkaline solution has a pH value in the range 7 < pH < 14.
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8. For hydrochloric acid, H  is about 5  10 2 moles per liter. Find the pH for
this strong acid to the nearest tenth.
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9. For milk of magnesia, H  is about 3.2  10 11 moles per liter. Find the pH of
milk of magnesia to the nearest tenth.
 
10. Find the pH of orange juice: H   3.16  10 4 moles per liter.
Mrs. Hayden
Magnitude of earthquakes
On the Richter scale, the magnitude, M, of an earthquake depends on the
amount of energy, E, released by the earthquake as follows:
log E = 11.8 + 1.5M
The amount of energy E, measured in ergs, is based on the amount of ground
motion recorded by a seismograph at a known distance from the epicenter of the
quake.
The logarithmic function for the Richter scale assigns very large numbers for the
amount of energy, E, to numbers that range from 1 to 9. A rating of 2 on the
Richter scale indicates the smallest tremor that can be detected. Destructive
earthquakes are those rated greater than 6 on the Richter scale.
-------------------------------------------------------------------------------------11. In 1976, an earthquake that released about 8  1019 ergs of energy occurred
in San Salvador, El Salvador. Find the magnitude, M, of this earthquake to the
nearest tenth.
12. One of the strongest earthquakes in recent history occurred in Mexico City in
1985. The amount of energy released by this earthquake was approximately
8.91 10 23 ergs. Find the magnitude, M, of this earthquake to the nearest tenth.
13. In 1989, an earthquake that measured 7.1 on the Richter scale occurred in
San Francisco, CA. Find the amount of energy, E, released by this earthquake.
14. On May 10, 1997, a light earthquake with a magnitude of 4.7 struck the
Calaveras Fault 10 miles east of San Jose, California. Find the amount of
energy, E, released by this earthquake.