One earthquake has MMS magnitude 3.1. If a second earthquake has 970 times as much energy (earth movement) as the first, find the magnitude of the second quake.
To find the magnitude of the second earthquake, we need to understand the relationship between earthquake magnitude and energy.
The magnitude of an earthquake measures the amount of energy released during the event. It is typically measured using the Moment Magnitude Scale (MMS), which is a logarithmic scale. This means that for each increase of 1 unit on the MMS, the energy released by the earthquake increases by a factor of 10.
In this case, we know that the first earthquake has a magnitude of 3.1. Let's call the magnitude of the second earthquake "M".
According to the given information, the second earthquake has 970 times the energy of the first earthquake. Since energy increases by a factor of 10 for each unit increase in magnitude, we can express the relationship between the magnitudes and energy as follows:
10^(M - 3.1) = 970
To find the value of M, we can take the logarithm of both sides of the equation. Let's use the base-10 logarithm (log):
log(10^(M - 3.1)) = log(970)
Using the logarithmic property log(a^b) = b * log(a), we can simplify the equation:
(M - 3.1) * log(10) = log(970)
Since log(10) = 1, we can further simplify the equation:
M - 3.1 = log(970)
Finally, we can solve for M by adding 3.1 to both sides of the equation:
M = 3.1 + log(970)
Using a calculator to evaluate log(970) (which is approximately 2.986), we can calculate the magnitude of the second earthquake:
M ≈ 3.1 + 2.986 ≈ 6.086
Therefore, the magnitude of the second earthquake is approximately 6.086 on the MMS.