Logarithm Applications
Date________________
Textbook Sections 7.3-7.5
1. The function 𝑦 = 20(.975)𝑥 models the intensity of sunlight beneath the surface of the
ocean. The output y represents the percent of surface sunlight intensity that reaches a
depth of x feet. The model is accurate from about 20 feet to about 600 feet beneath the
surface.
a) Find the percent of sunlight 50 feet beneath the surface of the ocean.
b) Find the percent of sunlight at a depth of 370 feet.
2. The half life of a radioactive substance is the time it takes for half of the material to
decay. Phosphorus-32 is used to study a plant’s use of fertilizer. It has a half-life of 14.3
days. Write the exponential decay function for a 50-mg sample. Find the amount of
phosphorus-32 remaining after 84 days.
3. In December 2004, an earthquake with magnitude 9.3 on the Richter scale hit off the
northwest coast of Sumatra. In March 2005, an earthquake with magnitude 8.7 on the
𝐼
Richter scale. The formula log 𝐼1 = 𝑀1 − 𝑀2 compares the intensity levels of earthquakes
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where I is the intensity level determined by a seismograph, and M is the magnitude on the
Richter scale. How many times more intense was the December earthquake than the March
earthquake?
Reference: Prentice Hall Algebra 2
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4. In 1812, an earthquake of magnitude 7.9 shook New Madrid, Missouri. Compare the
intensity of that earthquake to each earthquake below.
a) magnitude 7.7 in San Fransisco, California, in 1906
b) magnitude 9.5 in Valdivia, Chile, in 1960
c) magnitude 3.2 in Charlottesville, Virginia, in 2001
d) magnitude 6.9 in Kobe, Japan, in 1995
5. The pH of a substance equals − log[𝐻 + ], where [𝐻 + ] is the concentration of hydrogen
ions, and it ranges from 0 to 14. A pH of 7 is neutral. A level greater than 7 is basic, and a
level less than 7 is acidic. Find the concentration of hydrogen ions in seawater, if the pH
level of the seawater is 8.5.
6. The foreman of a construction team puts up a sound barrier that reduces the intensity
of the noise by 50%. By how many decibals is the noise reduced? Use the formula 𝐿 =
𝐼
10 log 𝐼 to measure loudness. (Hint: Find the difference between the expression for
0
loudness for intensity I and the expression for loudness for intensity 0.5I.
Reference: Prentice Hall Algebra 2
Page 2
7. The apparent brightness of stars is measured on a logarithmic scale called magnitude, in
which lower numbers mean brighter stars. The relationship between the ratio of apparent
brightness of two objects and the difference in their magnitudes is given by the formula
𝑏
𝑚2 − 𝑚1 = −2.5 log 𝑏2 , where m is the magnitude and b is the apparent brightness.
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a) How many times brighter is a magnitude 1.0 star than a magnitude 2.0 star?
b) The star Rigel has a magnitude of 0.12. How many times brighter is Capella (m =
0.1) than Rigel?
Reference: Prentice Hall Algebra 2
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