APPLICATIONS OF EXPONENTIAL
AND
LOGARITHMIC FUNCTIONS



EARTHQUAKE WORD PROBLEMS:



As with any word problem, the trick is convert a narrative statement or question to a mathematical statement.



Before we start, let's talk about earthquakes and how we measure their intensity.


In 1935 Charles Richter defined the magnitude of an earthquake to be

\begin{eqnarray*}M &=&\log \displaystyle \frac{I}{S}
\end{eqnarray*}

where I is the intensity of the earthquake (measured by the amplitude of a seismograph reading taken 100 km from the epicenter of the earthquake) and S is the intensity of a ''standard earthquake'' (whose amplitude is 1 micron =10-4 cm).


The magnitude of a standard earthquake is

\begin{eqnarray*}M &=&\log \displaystyle \frac{S}{S}=\log 1=0
\end{eqnarray*}


Richter studied many earthquakes that occurred between 1900 and 1950. The largest had magnitude of 8.9 on the Richter scale, and the smallest had magnitude 0. This corresponds to a ratio of intensities of 800,000,000, so the Richter scale provides more manageable numbers to work with.


Each number increase on the Richter scale indicates an intensity ten times stronger. For example, an earthquake of magnitude 6 is ten times stronger than an earthquake of magnitude 5. An earthquake of magnitude 7 is $10\times
10=100$ times strong than an earthquake of magnitude 5. An earthquake of magnitude 8 is $10\times 10\times 10=1000$ times stronger than an earthquake of magnitude 5.




Example 3: If one earthquake is 25 times as intense as another, how much larger is its magnitude on the Richter sclae?



Solution: Another of saying this is that one earthquake is 25 times as intense as another

\begin{eqnarray*}\displaystyle \frac{I_{1}}{I_{2}} &=&25
\end{eqnarray*}

where I1 is the intensity of the larger earthquake and I2 is the intensity of the smaller earthquake.


We are trying to determine the ratio of the larger magnitude M1 to the smaller magnitude I2 or M1-M2. The reason we are subtracting the magnitudes instead of dividing them is the question asked how much larger, not how many times larger.



Solve for I1 by multiplying both sides of the equation by I2.

\begin{eqnarray*}\displaystyle \frac{I_{1}}{I_{2}} &=&25 \\
&& \\
\displaystyl...
...}}\cdot I_{2} &=&25\cdot I_{2} \\
&& \\
I_{1} &=&25\cdot I_{2}
\end{eqnarray*}



We can write M1-M2 as $\log \displaystyle \frac{I_{1}}{S}-\log \displaystyle \frac{I_{2}}{S}$and we can write

\begin{eqnarray*}M_{1}-M_{2} &=&\log \displaystyle \frac{I_{1}}{S}-\log \display...
...style \frac{25\cdot I_{2}}{S}-\log \displaystyle \frac{I_{2}}{S}
\end{eqnarray*}

\begin{eqnarray*}M_{1}-M_{2} &=&\left( \log \displaystyle \frac{25\cdot I_{2}}{S...
...og 25\cdot I_{2}-\log S\right) -\left( \log I_{2}-\log S\right)
\end{eqnarray*}

\begin{eqnarray*}&=&\left( \log 25+\log I_{2}-\log S\right) -\left( \log I_{2}-\...
...ight)
\\
&& \\
&=&\log 25+\log I_{2}-\log S-\log I_{2}+\log S
\end{eqnarray*}

\begin{eqnarray*}&=&\log 25 \\
&& \\
&\approx &1.39794 \\
&& \\
M_{1}-M_{2} &\approx &1.4
\end{eqnarray*}



The larger earthquake had a magnitude 1.4 more on the Richter scale than the smaller earthquake.




Let's check our answer: Suppose the larger earthquake had a magnitude of 8.6 and the smaller earthquake had a magnitude of 8.6-1.4=7.2).

\begin{eqnarray*}L\arg er &:&8.6=\log \displaystyle \frac{I_{1}}{S} \\
&& \\
Smaller &:&7.2=\log \displaystyle \frac{I_{2}}{S}
\end{eqnarray*}



Convert both of these equations to exponential equations.


\begin{eqnarray*}L\arg er &:&10^{8.6}=\displaystyle \frac{I_{1}}{S} \\
&& \\
&&and \\
&& \\
S\cdot 10^{8.6} &=&I_{1}
\end{eqnarray*}

\begin{eqnarray*}Smaller &:&10^{7.2}=\displaystyle \frac{I_{2}}{S} \\
&& \\
S\cdot 10^{7.2} &=&I_{2}
\end{eqnarray*}

\begin{eqnarray*}\displaystyle \frac{I_{1}}{I_{2}} &=&\displaystyle \frac{S\cdot...
...10^{8.6}
}{10^{7.2}} \\
&& \\
&& \\
&=&10^{1.4}=25.1188643151
\end{eqnarray*}

\begin{eqnarray*}\displaystyle \frac{I_{1}}{I_{2}} &\approx &25
\end{eqnarray*}




Example 4: How much more intense is an earthquake of magnitude 6.5 on the Richter scale as one with a magnitude of 4.9?



Solution: The intensity (I) of each earthquake is different. Let I1 represent the intensity of the stronger earthquake and I2 represent the intensity of the weaker earthquake.

\begin{eqnarray*}First &:&6.5=\log \displaystyle \frac{I_{1}}{S} \\
&& \\
Second &:&4.9=\log \displaystyle \frac{I_{2}}{S}
\end{eqnarray*}



What you are looking for is the ratio of the intensities: $\displaystyle \frac{I_{1}}{
I_{2}}.$ So our task is to isolate this ratio from the above given information using the rules of logarithms.


\begin{eqnarray*}\log \displaystyle \frac{I_{1}}{S}-\log \displaystyle \frac{I_{...
...\log I_{1}-\log S\right) -\left( \log I_{2}-\log S\right) &=&1.6
\end{eqnarray*}

\begin{eqnarray*}\log I_{1}-\log S-\log I_{2}+\log S &=&1.6 \\
&& \\
\log I_{1...
...2} &=&1.6\\
&&\\
\log \displaystyle \frac{I_{1}}{I_{2}} &=&1.6
\end{eqnarray*}



Convert the logarithmic equation to an exponential equation.


\begin{eqnarray*}\log \displaystyle \frac{I_{1}}{I_{2}} &=&1.6 \\
&& \\
10^{1....
...{2}}\\
&&\\
\displaystyle \frac{I_{1}}{I_{2}} &=&39.8107170554
\end{eqnarray*}

\begin{eqnarray*}\displaystyle \frac{I_{1}}{I_{2}} &\approx &40 \\
&& \\
&&or \\
&& \\
I_{1} &=&40\cdot I_{2}
\end{eqnarray*}



The stronger earthquake was 40 times as intense as the weaker earthquake.





If you would like to work another example, click on example.


If you would like to test your knowledge by working some problems, click on problem.


If you would like to go back to the table of contents, click on contents.

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Author: Nancy Marcus

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