Introduction to Logarithmic Functions

Photo of the aftermath of the earthquake in Japan with a focus on the Japanese flag.

Devastation of the earthquake which happened on March 11, 2011 in Honshu, Japan. (credit: Daniel Pierce)

In [latex]2010[/latex], a major earthquake struck Haiti, destroying or damaging over [latex]285,000[/latex] homes.[1] One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over [latex]332,000[/latex] buildings,[2] like those shown in the picture above. Even though both caused substantial damage, the earthquake in [latex]2011[/latex] was [latex]100[/latex] times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a [latex]7.0[/latex] on the Richter Scale[3] whereas the Japanese earthquake registered a [latex]9.0[/latex].[4]

The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude [latex]4[/latex]. It is [latex]{10}^{8 - 4}={10}^{4}=10,000[/latex] times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten function upon which it depends.