To compare the intensities of the two earthquakes based on their magnitudes, we follow the relationship given:
[tex]$
M = \log \left(\frac{I}{I_0}\right)
$[/tex]
where [tex]\( M \)[/tex] is the magnitude, [tex]\( I \)[/tex] is the intensity, and [tex]\( I_0 \)[/tex] is the reference intensity.
First, recall that the logarithmic relationship can be rearranged to express intensity [tex]\( I \)[/tex] in terms of magnitude [tex]\( M \)[/tex]:
[tex]$
I = I_0 \cdot 10^M
$[/tex]
Since [tex]\( I_0 \)[/tex] is the smallest detectable seismic activity and can be considered as 1 (for simplicity in our calculations), we have:
[tex]$
I = 10^M
$[/tex]
### Step-by-Step Solution:
1. Calculate the intensity for the first earthquake:
- Magnitude of the first earthquake ([tex]\( M_1 \)[/tex]) = 8.6
Using the formula:
[tex]$
I_1 = 10^{8.6}
$[/tex]
From the given information, we know:
[tex]$
I_1 \approx 398107170.5534969
$[/tex]
2. Calculate the intensity for the second earthquake:
- Magnitude of the second earthquake ([tex]\( M_2 \)[/tex]) = 8.2
Using the formula:
[tex]$
I_2 = 10^{8.2}
$[/tex]
From the given information, we know:
[tex]$
I_2 \approx 158489319.2461111
$[/tex]
3. Calculate the factor by which the intensity of the first earthquake is greater than the intensity of the second earthquake:
We find this factor by dividing the intensity of the first earthquake by the intensity of the second earthquake:
[tex]$
\text{Factor} = \frac{I_1}{I_2}
$[/tex]
Using the provided numerical values:
[tex]$
\text{Factor} \approx \frac{398107170.5534969}{158489319.2461111} \approx 2.511886431509582
$[/tex]
### Conclusion:
The intensity of the first earthquake (magnitude 8.6) was approximately 2.51 times greater than the intensity of the second earthquake (magnitude 8.2).
