The magnitude of an earthquake is measured relative to the strength of a "standard" earthquake, whose seismic waves are of size $W_{0}$. The magnitude, $M$, of an earthquake with seismic waves of size $W$ is defined to be $M=\log \left(\frac{W}{W_{0}}\right)$. The value of $M$ is called the Richter scale rating of the strength of an earthquake. (a) Let $M_{1}$ and $M_{2}$ represent the magnitude of two earthquakes whose seismic waves are of sizes $W_{1}$ and $W_{2}$, respectively. Using log properties, find a simplified formula for the difference $M_{2}-M_{1}$ in terms of $W_{1}$ and $W_{2}$. \[ M_{2}-M_{1}= \] (b) The April 2017 earthquake 69 km SSE of Adak, Alaska, had a rating of 5.1 on the Richter scale ${ }^{1}$. How many times larger than the Alaska earthquake were the seismic waves in the 1883 Krakatoa eruption, ${ }^{2}$ which released the energy equivalent of a magnitude 8.5 earthquake? Give your answer to the nearest integer.

Step 1 ：The magnitude of an earthquake is measured relative to the strength of a 'standard' earthquake, whose seismic waves are of size \(W_{0}\). The magnitude, \(M\), of an earthquake with seismic waves of size \(W\) is defined to be \(M=\log \left(\frac{W}{W_{0}}\right)\). The value of \(M\) is called the Richter scale rating of the strength of an earthquake.

Step 2 ：Let \(M_{1}\) and \(M_{2}\) represent the magnitude of two earthquakes whose seismic waves are of sizes \(W_{1}\) and \(W_{2}\), respectively. We know that \(M_{1}=\log \left(\frac{W_{1}}{W_{0}}\right)\) and \(M_{2}=\log \left(\frac{W_{2}}{W_{0}}\right)\). So, \(M_{2}-M_{1}=\log \left(\frac{W_{2}}{W_{0}}\right)-\log \left(\frac{W_{1}}{W_{0}}\right)\). Using the properties of logarithms, we can simplify this expression to \(\log \left(\frac{W_{2}}{W_{1}}\right)\).

Step 3 ：The April 2017 earthquake 69 km SSE of Adak, Alaska, had a rating of 5.1 on the Richter scale. The 1883 Krakatoa eruption released the energy equivalent of a magnitude 8.5 earthquake. So, the difference in their magnitudes is \(8.5-5.1=3.4\). Using the formula we derived, we can find the ratio of the seismic waves' sizes.

Step 4 ：The seismic waves of the Krakatoa eruption were approximately \(\boxed{30}\) times larger than those of the Alaska earthquake.