First, let's understand the given formula for the magnitude of an earthquake:
[tex]\[ M = \log \frac{1}{S} \][/tex]
where:
- [tex]\( M \)[/tex] is the magnitude of the earthquake
- [tex]\( S \)[/tex] is the intensity of a standard, barely detectable earthquake
Now, we need to determine the magnitude of an earthquake that is 10 times more intense than a standard earthquake.
Given that the intensity of the earthquake ([tex]\( I \)[/tex]) is 10 times the intensity of a standard earthquake ([tex]\( S \)[/tex]), we can express this relationship as:
[tex]\[ I = 10 \times S \][/tex]
We need to find the appropriate equation for the magnitude of this earthquake.
Starting from the definition of magnitude [tex]\( M = \log \frac{1}{I} \)[/tex], let's substitute [tex]\( I = 10 \times S \)[/tex]:
[tex]\[ M = \log \frac{1}{10 \times S} \][/tex]
Now, let's write this equation in a simpler form:
[tex]\[ M = \log \frac{1}{10S} \][/tex]
That’s it! We have derived the equation.
The correct equation that represents the magnitude of an earthquake that is 10 times more intense than a standard earthquake is:
[tex]\[ M = \log \frac{1}{10S} \][/tex]
This matches one of the given options. Therefore, the correct answer is:
[tex]\[ \boxed{M = \log \frac{1}{10 S}} \][/tex]
