This problem uses the Richter scale for the strength of an earthquake. The strength, $W$, of the seismic waves of an earthquake are compared to the strength, $W_{0}$, of the seismic waves of a standard earthquake. The Richter scale rating, $M$, is \[ M=\log \left(\frac{W}{W_{0}}\right) \] How many times larger are the seismic waves of an earthquake with a rating of 8.1 on the Richter scale, than the seismic waves of an earthquake with a rating of 4.2 ? Round your answer to the nearest integer. The seismic waves of an earthquake with a rating of 8.1 are $\mathbf{i}$ times larger than the seismic waves of an earthquake with a rating of 4.2 .

Step 1 ：Given the Richter scale ratings of two earthquakes as 8.1 and 4.2 respectively.

Step 2 ：The strength, $W$, of the seismic waves of an earthquake are compared to the strength, $W_{0}$, of the seismic waves of a standard earthquake. The Richter scale rating, $M$, is given by the formula $M=\log \left(\frac{W}{W_{0}}\right)$

Step 3 ：To calculate the ratio of the seismic waves of the two earthquakes, subtract the Richter scale rating of the earthquake with a rating of 4.2 from the Richter scale rating of the earthquake with a rating of 8.1, and then raise 10 to the power of this difference. This will give us the ratio of the seismic waves of the two earthquakes.

Step 4 ：Performing the calculation gives a ratio of approximately 7943.282347242805

Step 5 ：Rounding this ratio to the nearest integer gives a final answer of 7943

Step 6 ：Final Answer: The seismic waves of an earthquake with a rating of 8.1 are \(\boxed{7943}\) times larger than the seismic waves of an earthquake with a rating of 4.2.