Putting It Together: Functions and Graphs

The Richter Scale for Earthquakes

In 1935, Charles Richter developed a scale (now known as the Richter scale) to measure the magnitude of an earthquake. The scale is a base-10 logarithmic scale, and it can be described as follows: Consider one earthquake with magnitude [latex]R_1[/latex] on the Richter scale and a second earthquake with magnitude [latex]R_2[/latex] on the Richter scale. Suppose [latex]R_1 > R_2[/latex], which means the earthquake of magnitude [latex]R_1[/latex] is stronger, but how much stronger is it than the other earthquake? A way of measuring the intensity of an earthquake is by using a seismograph to measure the amplitude of the earthquake waves. If [latex]A_1[/latex] is the amplitude measured for the first earthquake and [latex]A_2[/latex] is the amplitude measured for the second earthquake, then the amplitudes and magnitudes of the two earthquakes satisfy the following equation:

[latex]R_1 - R_2 = \log_{10}\left(\frac{A_1}{A_2}\right)[/latex]

Consider an earthquake that measures 8 on the Richter scale and an earthquake that measures 7 on the Richter scale. Then,

[latex]8-7=\log_{10}\left(\frac{A_1}{A_2}\right)[/latex]

Therefore,

[latex]\log_{10}\left(\frac{A_1}{A_2}\right)=1[/latex],

which implies [latex]A_1 / A_2 = 10[/latex] or [latex]A_1 = 10A_2[/latex]. Since [latex]A_1[/latex] is 10 times the size of [latex]A_2[/latex], we say that the first earthquake is 10 times as intense as the second earthquake. On the other hand, if one earthquake measures 8 on the Richter scale and another measures 6, then the relative intensity of the two earthquakes satisfies the equation

[latex]\log_{10}\left(\frac{A_1}{A_2}\right)=8-6=2[/latex]

Therefore, [latex]A_1=100A_2[/latex]. That is, the first earthquake is 100 times more intense than the second earthquake.

How can we use logarithmic functions to compare the relative severity of the magnitude 9 earthquake in Japan in 2011 with the magnitude 7.3 earthquake in Haiti in 2010?

To compare the Japan and Haiti earthquakes, we can use an equation presented earlier:

[latex]9-7.3=\log_{10}\left(\frac{A_1}{A_2}\right)[/latex]

Therefore, [latex]\frac{A_1}{A_2}=10^{1.7}[/latex], and we conclude that the earthquake in Japan was approximately 50 times more intense than the earthquake in Haiti.