To address the problem, let's analyze the given formula for the magnitude [tex]\( M \)[/tex] of an earthquake:
[tex]\[ M = \log \frac{I}{S} \][/tex]
where:
- [tex]\( I \)[/tex] is the intensity of the earthquake.
- [tex]\( S \)[/tex] is the intensity of a standard earthquake.
1. Standard Earthquake Magnitude:
For a standard earthquake, the intensity [tex]\( I \)[/tex] is equal to [tex]\( S \)[/tex]. Therefore, the magnitude [tex]\( M \)[/tex] of a standard earthquake is given by:
[tex]\[ M = \log \frac{S}{S} = \log 1 = 0 \][/tex]
2. Earthquake 100 Times More Intense:
We need to find the magnitude [tex]\( M \)[/tex] for an earthquake that is 100 times more intense than a standard earthquake. This means the intensity [tex]\( I \)[/tex] is [tex]\( 100 \times S \)[/tex].
3. Substitute Intensity:
Substitute [tex]\( I = 100S \)[/tex] into the magnitude formula:
[tex]\[ M = \log \frac{100S}{S} \][/tex]
4. Simplify the Expression:
Simplify the fraction inside the logarithm:
[tex]\[ M = \log \frac{100S}{S} = \log 100 \][/tex]
5. Logarithmic Calculation:
The logarithm of 100 to the base 10 is:
[tex]\[ \log 100 = 2 \][/tex]
Thus, the magnitude [tex]\( M \)[/tex] of an earthquake that is 100 times more intense than a standard earthquake is:
[tex]\[ M = 2 \][/tex]
Therefore, the correct equation that represents the magnitude of an earthquake that is 100 times more intense than a standard earthquake is:
[tex]\[ M = \log \frac{100S}{S} \][/tex]
So, the correct option among the provided choices is:
[tex]\[ M = \log \frac{100S}{S} \][/tex]
