To determine the correct equation for the magnitude of an earthquake that is 10 times more intense than a standard earthquake, let's use the formula for the magnitude of an earthquake:
[tex]\[ M = \log \left( \frac{I}{S} \right) \][/tex]
Where:
- [tex]\( M \)[/tex] is the magnitude.
- [tex]\( I \)[/tex] is the intensity of the earthquake.
- [tex]\( S \)[/tex] is the intensity of a "standard" earthquake.
Given that the intensity [tex]\( I \)[/tex] of the earthquake is 10 times the intensity of a standard earthquake [tex]\( S \)[/tex], we can write:
[tex]\[ I = 10S \][/tex]
Using the formula for the magnitude:
[tex]\[ M = \log \left( \frac{I}{S} \right) \][/tex]
Substitute [tex]\( I = 10S \)[/tex]:
[tex]\[ M = \log \left( \frac{10S}{S} \right) \][/tex]
Simplify the fraction:
[tex]\[ M = \log (10) \][/tex]
Therefore, the equation that represents the magnitude of an earthquake that is 10 times more intense than a standard earthquake is:
[tex]\[ M = \log (10) \][/tex]
Hence, the answer is not among the provided options. However, if we consider the provided options, the closest interpretation could be:
[tex]\[ M = \log (10S) \][/tex]
Although this is not a perfect match to our simplified equation, it is the closest representation among the four choices given. Thus, the best matching answer among the provided options would be:
[tex]\[ M = \log (10S) \][/tex]
