To determine the equation representing the magnitude of an earthquake that is 100 times more intense than a standard earthquake, we need to consider the given options and analyze which best fits this situation.
Let's start by recalling the conceptual relationship of earthquake intensity and magnitude in general terms. The Richter scale, commonly used for measuring earthquake magnitude, defines magnitude [tex]\( M \)[/tex] by the equation:
[tex]\[ M = \log \left( \frac{I}{S} \right) \][/tex]
where:
- [tex]\( I \)[/tex] is the intensity of the earthquake,
- [tex]\( S \)[/tex] is a standard reference intensity.
For an earthquake that is 100 times more intense than a standard earthquake, we denote the intensity of this more intense earthquake as [tex]\( 100S \)[/tex], where [tex]\( S \)[/tex] is the intensity of the standard earthquake.
Given this, we substitute [tex]\( 100S \)[/tex] into the generic magnitude equation:
[tex]\[ M = \log \left( \frac{100S}{S} \right) \][/tex]
This simplifies to:
[tex]\[ M = \log (100) \][/tex]
Expressing this in terms of the given options, we see that the correct equation is:
[tex]\[ M = \log (100S) \][/tex]
So, the magnitude [tex]\( M \)[/tex] is:
[tex]\[ M = \log (100 S) \][/tex]
