A formula for calculating the magnitude of an earthquake is  M=23log(EE0)
  that uses the common (base 10) logarithm. This is called the Moment Magnitude Scale (MMS), an alternative to the more well known Richter Scale. One earthquake has magnitude  3.9
  on the MMS. If a second earthquake has  650
  times as much energy as the first, find the magnitude of the second quake.

We are given the Moment Magnitude Scale (MMS) formula:

\[
M = 2/3 \log\left(\frac{E}{E_0}\right)
\]

Where:
- \( M \) is the magnitude of the earthquake,
- \( E \) is the energy released by the earthquake,
- \( E_0 \) is a reference energy.

We are told:
- The first earthquake has a magnitude of \( M_1 = 3.9 \).
- The second earthquake releases \( 650 \) times as much energy as the first earthquake.

We want to find the magnitude of the second earthquake, \( M_2 \).

### Step 1: Relationship between the energy and magnitudes
We know that for two earthquakes:
\[
M_1 = \frac{2}{3} \log\left(\frac{E_1}{E_0}\right)
\]
\[
M_2 = \frac{2}{3} \log\left(\frac{E_2}{E_0}\right)
\]

Also, we are given that:
\[
E_2 = 650 \cdot E_1
\]

### Step 2: Express the ratio of energies in terms of magnitudes
Using the energy ratio, we can relate the magnitudes of the two earthquakes. First, express the energy ratio in terms of magnitudes:
\[
M_2 - M_1 = \frac{2}{3} \log\left(\frac{E_2}{E_0}\right) - \frac{2}{3} \log\left(\frac{E_1}{E_0}\right)
\]

Using the logarithmic property \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \), we simplify:
\[
M_2 - M_1 = \frac{2}{3} \log\left(\frac{E_2}{E_1}\right)
\]
Since \( E_2 = 650 \cdot E_1 \), we have:
\[
M_2 - M_1 = \frac{2}{3} \log(650)
\]

### Step 3: Solve for \( M_2 \)
Now, calculate the value of \( \log(650) \):
\[
\log(650) \approx 2.8129
\]

Thus:
\[
M_2 - 3.9 = \frac{2}{3} \times 2.8129
\]
\[
M_2 - 3.9 \approx 1.8753
\]
\[
M_2 \approx 3.9 + 1.8753
\]
\[
M_2 \approx 5.7753
\]

### Conclusion:
The magnitude of the second earthquake is approximately \( 5.78 \) on the Moment Magnitude Scale (MMS).

---

Would you like any further details or explanations on this? Here are some related questions to expand your understanding:

1. How does the Moment Magnitude Scale differ from the Richter Scale in terms of calculation?
2. What are the limitations of the MMS for very large earthquakes?
3. Can you explain why logarithmic scales are used for earthquake magnitudes?
4. How is energy release related to ground motion in an earthquake?
5. What are some examples of notable earthquakes and their magnitudes on the MMS?

**Tip:** Logarithmic scales are useful for representing quantities that vary over a large range, as small changes in magnitude correspond to large changes in energy.

A formula for calculating the magnitude of an earthquake is M=2/3log(E/E0) that uses the common (base 10) logarithm. This is called the Moment Magnitude Scale (MMS), an alternative to the more well-known Richter Scale. One earthquake has magnitude 3.9 on the MMS. If a second earthquake has 650 times as much energy as the first, find the magnitude of the second quake.

Learn how to calculate the magnitude of an earthquake using the Moment Magnitude Scale (MMS). This example demonstrates how to solve for the magnitude of a second earthquake that releases 650 times more energy than the first, using logarithmic properties. Ideal for students in Grades 10-12.

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