To determine how many times more intense the San Francisco earthquake was compared to the South American earthquake, we need to understand the relationship between the magnitudes of the two earthquakes.
The given formula for the magnitude of an earthquake is:
[tex]\[ R = \log_{10}\left(\frac{I_c}{I_n}\right) \][/tex]
Here, [tex]\( I_c \)[/tex] is the intensity of the earthquake and [tex]\( I_n \)[/tex] is the intensity of a standard earthquake.
To find the intensity ratio between the San Francisco earthquake ([tex]\(R_{sf} = 8.3\)[/tex]) and the South American earthquake ([tex]\(R_{sa} = 5.3\)[/tex]), we follow these steps:
1. Express the magnitudes in terms of their intensities:
[tex]\[ R_{sf} = \log_{10}\left(\frac{I_{sf}}{I_n}\right) \][/tex]
[tex]\[ R_{sa} = \log_{10}\left(\frac{I_{sa}}{I_n}\right) \][/tex]
2. Subtract the two equations to find the logarithm of the intensity ratio:
[tex]\[ R_{sf} - R_{sa} = \log_{10}\left(\frac{I_{sf}}{I_n}\right) - \log_{10}\left(\frac{I_{sa}}{I_n}\right) \][/tex]
[tex]\[ 8.3 - 5.3 = \log_{10}\left(\frac{I_{sf}}{I_n}\right) - \log_{10}\left(\frac{I_{sa}}{I_n}\right) \][/tex]
[tex]\[ 3.0 = \log_{10}\left(\frac{I_{sf}}{I_n} \cdot \frac{I_n}{I_{sa}}\right) \][/tex]
[tex]\[ 3.0 = \log_{10}\left(\frac{I_{sf}}{I_{sa}}\right) \][/tex]
3. Use the properties of logarithms to solve for the intensity ratio:
[tex]\[ 10^{3.0} = \frac{I_{sf}}{I_{sa}} \][/tex]
4. Calculate the intensity ratio:
[tex]\[ \frac{I_{sf}}{I_{sa}} = 10^3 = 1000 \][/tex]
Thus, the San Francisco earthquake was 1000 times more intense than the South American earthquake.
Rounded to two decimal places, the San Francisco earthquake was 1000.00 times more intense than the South American earthquake.
Therefore, the San Francisco earthquake was [tex]\(\boxed{1000.00}\)[/tex] times more intense than the South American earthquake.
