A formula for calculating the magnitude of an earthquake is \( M=\frac{2}{3} \log \left(\frac{E}{E_{0}}\right) \) 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 750 times as much energy as the first, find the magnitude of the second quake. Round to the nearest hundredth. The magnitude of the second earthquake was Number

To find the magnitude of the second earthquake, we start with the formula given: \[ M = \frac{2}{3} \log \left(\frac{E}{E_{0}}\right) \] Given that the second earthquake has 750 times the energy of the first, we denote the energy of the first earthquake as \( E_{1} \) and the second as \( E_{2} = 750 \cdot E_{1} \). Using the formula, we know: \[ M_1 = 3.9 = \frac{2}{3} \log \left(\frac{E_{1}}{E_{0}}\right) \] Now let's find \( M_2 \): \[ M_2 = \frac{2}{3} \log \left(\frac{E_{2}}{E_{0}}\right) = \frac{2}{3} \log \left(\frac{750 \cdot E_{1}}{E_{0}}\right) \] This can be broken down as: \[ M_2 = \frac{2}{3} \log \left(750\right) + \frac{2}{3} \log \left(\frac{E_{1}}{E_{0}}\right) \] Substituting \( M_1 \) into the equation: \[ M_2 = \frac{2}{3} \log \left(750\right) + M_1 \] Now calculate \( \log(750) \): \[ \log(750) \approx 2.875 \] Putting it all together: \[ M_2 = \frac{2}{3} \times 2.875 + 3.9 \approx 1.9167 + 3.9 \approx 5.8167 \] Rounding to the nearest hundredth gives: \[ \text{The magnitude of the second earthquake was } 5.82. \]