To determine which equation represents the magnitude [tex]\( M \)[/tex] of an earthquake that is 10 times more intense than a standard earthquake, we need to carefully analyze the given formula for the magnitude of an earthquake:
[tex]\[ M = \log \frac{1}{S} \][/tex]
Here the parameter [tex]\( I \)[/tex] is the intensity of the earthquake and [tex]\( S \)[/tex] is the standard intensity.
1. Adjustment for Increased Intensity:
- Given that the earthquake intensity is 10 times more than the standard earthquake, the intensity [tex]\( I \)[/tex] of the new earthquake would be:
[tex]\[ I = 10 \cdot S \][/tex]
2. Substitute Intensity into Magnitude Formula:
- Substitute [tex]\( I = 10S \)[/tex] into the original formula:
[tex]\[ M = \log \frac{1}{I} \][/tex]
[tex]\[ M = \log \frac{1}{10S} \][/tex]
Therefore, based on the above steps and substitution, 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]
The other options can be evaluated but they do not fit the requirement set by the standard formula or the given conditions:
- [tex]\( M = \log (10S) \)[/tex] would imply we are calculating the logarithm of the intensity directly, not its reciprocal.
- [tex]\( M = \log \frac{10S}{S} \)[/tex] simplifies incorrectly to [tex]\( M = \log 10 \)[/tex].
- [tex]\( M = \log \frac{10}{S} \)[/tex] represents another manipulation but not the one derived correctly from the given conditions.
Thus, the correct answer is:
[tex]\[
M = \log \frac{1}{10S}
\][/tex]
