Introduction: Logarithmic Functions

Figure 1. Devastation of March 11, 2011 earthquake in Honshu, Japan. (credit: Daniel Pierce)

In [latex]2010[/latex], a major earthquake struck Haiti, destroying or damaging over [latex]285,000[/latex] homes.[footnote]http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/#summary. Accessed [latex]3/4/2013[/latex].[/footnote] One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over [latex]332,000[/latex] buildings,[footnote]http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#summary. Accessed [latex]3/4/2013[/latex].[/footnote] 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[footnote]http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/. Accessed [latex]3/4/2013[/latex].[/footnote] whereas the Japanese earthquake registered a [latex]9.0[/latex].[footnote]http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#details. Accessed [latex]3/4/2013[/latex].[/footnote]

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.

The learning objectives for this lesson include:

• Convert from logarithmic to exponential form
• Convert from exponential to logarithmic form
• Evaluate logarithms
• Use common logarithms
• Use natural logarithms

The learning activities for this lesson include:

• Read: Composite and Inverse Functions
• Self-Check: Composite and Inverse Functions
• Read: Define Logarithmic Functions
• Self-Check: Define Logarithmic Functions
• Read: Evaluate Logarithms
• Self-Check:Evaluate Logarithms
• Read: Graphs of Logarithmic Functions
• Self-Check: Graphs of Logarithmic Functions

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