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 7.9 on the Richter scale, than the seismic waves of an earthquake with a rating of 5.1 ? Round your answer to the nearest integer. The seismic waves of an earthquake with a rating of 7.9 are earthquake with a rating of 5.1 . times larger than the seismic waves of an i times larger than the seismic waves of an

Step 1 ：Given two Richter scale ratings, 7.9 and 5.1.

Step 2 ：Express the strength of the seismic waves of the two earthquakes in terms of \(W_0\) using the formula for the Richter scale rating: \[M=\log \left(\frac{W}{W_{0}}\right)\]

Step 3 ：For an earthquake with a rating of 7.9, we have \[W_1 = 10^{7.9} \times W_0 = 79432823.47242822 \times W_0\]

Step 4 ：For an earthquake with a rating of 5.1, we have \[W_2 = 10^{5.1} \times W_0 = 125892.54117941661 \times W_0\]

Step 5 ：Divide the strength of the seismic waves of the earthquake with a rating of 7.9 by the strength of the seismic waves of the earthquake with a rating of 5.1 to find out how many times larger the former is than the latter: \[\frac{W_1}{W_2} = \frac{79432823.47242822 \times W_0}{125892.54117941661 \times W_0} = 631\]

Step 6 ：Final Answer: The seismic waves of an earthquake with a rating of 7.9 on the Richter scale are \(\boxed{631}\) times larger than the seismic waves of an earthquake with a rating of 5.1.