Semester Project

Introduction

Today, logarithms are used to describe the world around us. One common application of logarithms is with earthquakes and the Richter scale. Earthquakes are assigned a number on the Richter scale based on their intensities, but what exactly do those numbers mean?

Let's consider two earthquakes. Right after the world shut down due to the Covid-19 pandemic in March 2020, a 5.7 magnitude earthquake shook the homes of northern Utah residents (Wikimedia, 2021). Now, let's compare that with the 8.2 magnitude earthquake in Alaska in July 2021 (USGS, 2021).

We are used to interval scales rather than logarithmic scales, so we often think an 8.2 earthquake is not that much larger than 5.7. To us, \(\frac{8.2}{5.7}=1.43\); therefore, the earthquake in Alaska was only about one and a half times larger than the one experienced in Utah. However, an appropriate interpretation of the logarithmic nature of the Richter scale will show us why the roads in Alaska were destroyed (see the image to the right) after an 8.2 magnitude earthquake and why some Utah residents slept through the 5.7 magnitude earthquake.

The equation that models the Richter scale is as follows:

We can plug in the known magnitudes of the earthquakes to figure out the intensities:

Let's rewrite the above equation in exponent form:

Amazing! The earthquake in Alaska was over 300 times larger in intensity than the quake felt in Utah!