Name: __________________
Date: ______________
Unit 2: Families of Functions
Applications of Logarithms
________________________________________________________________
Directions: Answer all questions to the best of your ability. This
assignment is due at the start of class on Friday, January 12. You may use
extra paper if necessary. The assessment criteria is attached. This is an
individual assignment—you may ask me for help, but no one else. Sign the
principled statement at the end of the document before you turn it in.
The severity of an earthquake is measured and recorded as a magnitude, 𝑀,
often referred to as its reading on the Richter scale. The Richter scale starts at 0
and goes as high as 9, but earthquakes are usually measured between 2 and 9.
For instance, an earthquake in New Zealand in 2011 had a magnitude 𝑀 of 6.1.
A recent earthquake in Italy—about a year ago—had a magnitude of 5.7. An
earthquake in Nepal in 2015 that killed 9,000 people had a magnitude of 7.8.
The magnitude is defined as a function of
A, the maximum amplitude of horizontal
displacement of the earth’s surface, at a
distance of 100 km from the epicenter of
the earthquake. The picture at right shows
how a seismometer, an instrument used
to measure earthquakes, records
vibrations in the earth’s surface.
Magnitude and amplitude are related by
the following equation:
𝑀 = log
𝐴
𝐴0
where 𝐴0 = 1 𝜇𝑚 = 1 × 10−6 𝑚 is a constant of displacement chosen as a
reference by Charles Richter, the inventor of the Richter scale, in 1935.
1.) Calculate the magnitude 𝑀 of an earthquake for the following displacements
and then and use the table on the following page to describe what it might have
felt like if you experienced these earthquakes.
a.) 𝐴 = 100 𝜇m
b.) 𝐴 = 10 m (convert to 𝜇m first)
c.) 𝐴 = 30 cm (convert to 𝜇m first)
𝐴
2.) Rearrange the logarithmic equation 𝑀 = log 𝐴 into an equivalent exponential
0
equation.
3.) Using your equation from (2), calculate A, the surface displacement, for the
following earthquakes. Use 𝑨𝟎 = 𝟏𝟎−𝟔 when you solve for A. The units for 𝐴
are meters.
a.) New Zealand, 2011, 𝑀 = 6.1
b.) Amatrice, Italy, 2017, 𝑀 = 5.7
c.) Chile, 1960, 𝑀 = 9.5
4.) In 1906, an earthquake in San Francisco registered 8.3 on the Richter scale
(and resulted in much devastation and the deaths of over 3000 people). In the
same year, another earthquake was recorded in South America that was four
times stronger. What was the magnitude of the earthquake in South America?
Hints: (1) You’ll need two equations since there are 2 earthquakes.
(2) Let 𝑀𝑆𝐹 = 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑆𝑎𝑛 𝐹𝑟𝑎𝑛𝑐𝑖𝑠𝑐𝑜 𝑒𝑎𝑟𝑡ℎ𝑞𝑢𝑎𝑘𝑒 and let 𝑀𝑆𝐴 =
𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑆𝑜𝑢𝑡ℎ 𝐴𝑚𝑒𝑟𝑖𝑐𝑎 𝑒𝑎𝑟𝑡ℎ𝑞𝑢𝑎𝑘𝑒
(3) Since the South America earthquake was 4 times stronger than the
San Francisco earthquake, 𝐴𝑆𝐴 = 4𝐴𝑆𝐹 , where 𝐴𝑆𝐴 is the amplitude of the South
America earthquake and 𝐴𝑆𝐹 is the amplitude of the San Francisco earthquake.
(4) Use (3) to substitute and then use the quotient law of logarithms to
𝑎
rewrite log10 𝑏 as two logarithms. Solve.
5.) A recent earthquake in Italy measured 5.7 on the Richter scale. How many
times more intense was the San Francisco earthquake described in problem 4?
Hints: (1) You will need two equations, one for each earthquake, like in
problem 4, but this time you have the magnitudes (5.7 and 8.3), and you’re
interested in solving for the amplitudes. Remember that 𝐴0 = 10−6 m.
(2) You can solve for the amplitudes in both equations and then
compare them.
(3) Find the ratio
Francisco earthquake was.
𝐴𝑆𝐹
𝐴𝐼
to determine how much stronger the San