One Problem at a Time

Application of Logarithm (Earthquake Intensity)

Problem 207: In 1906, San Francisco experienced an intense earthquake with a magnitude of 7.8 on the Richter scale. Over 80% of the city was destroyed by the resulting fires. In 2014, Los Angeles experienced a moderate earthquake that measured 5.1 on the Richter scale and caused $108 million dollars of damage. Compare the intensities of the two earthquakes.

Solution: The magnitude of an earthquake is measured by \log I, where I is the intensity of its shock wave.

For the 1906 earthquake, the intensity is given by

(1)   \begin{equation*} 7.8 = \log I \quad \Rightarrow \quad 10^{7.8} = I.\end{equation*}

For the 2014 earthquake, the intensity is given by

(2)   \begin{equation*} 5.1 = \log J \quad \Rightarrow \quad 10^{5.1} = J.\end{equation*}

Then we can take their ratio to compare their intensity. Thus

(3)   \begin{equation*}\frac{10^{7.8}}{10^{5.1}} = 10^{2.7} = 501. \end{equation*}

This means that the intensity of the 1905 earthquake in San Francisco was 501 more than the earthquake in 2014 in Los Angeles.

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