To find the equation that represents the magnitude of an earthquake that is 100 times more intense than a standard earthquake, let's follow the steps involved:
1. Understand the Standard Magnitude Formula:
The Richter scale is a logarithmic scale used to express the magnitude of an earthquake. The formula for magnitude [tex]\( M \)[/tex] in terms of intensity [tex]\( I \)[/tex] and a standard reference intensity [tex]\( S \)[/tex] is:
[tex]\[
M = \log \left( \frac{I}{S} \right)
\][/tex]
2. Intensity Relation:
We are given that the earthquake in question is 100 times more intense than a standard earthquake. Therefore, let the intensity of the standard earthquake be [tex]\( S \)[/tex]. The intensity of the earthquake in question is [tex]\( 100S \)[/tex].
3. Substitute Intensities:
Substitute [tex]\( 100S \)[/tex] for [tex]\( I \)[/tex] in the standard magnitude formula:
[tex]\[
M = \log \left( \frac{100S}{S} \right)
\][/tex]
4. Simplify the Equation:
Simplify the fraction inside the logarithm:
[tex]\[
\frac{100S}{S} = 100
\][/tex]
This simplifies the equation to:
[tex]\[
M = \log (100)
\][/tex]
5. Conclusion:
Therefore, the correct equation that represents the magnitude of an earthquake that is 100 times more intense than a standard earthquake is:
[tex]\[
M = \log (100 S)
\][/tex]
From the given options, the correct equation is:
[tex]\[
M = \log (100 S)
\][/tex]
