Step 1 :Given that the intensity of the earthquake is \(10^{8.16} \cdot I_{0}\), we can substitute this into the formula for the Richter scale magnitude.
Step 2 :So, \(R = \log \frac{1}{10^{8.16} \cdot I_{0}}\)
Step 3 :We know that \(\log \frac{1}{a} = -\log a\), so we can simplify the equation to: \(R = -\log (10^{8.16} \cdot I_{0})\)
Step 4 :We can separate the logarithm into two parts: \(R = -(\log 10^{8.16} + \log I_{0})\)
Step 5 :We know that \(\log 10^{a} = a\), so we can simplify the equation to: \(R = -(8.16 + \log I_{0})\)
Step 6 :Since \(I_{0}\) is a constant, \(\log I_{0}\) is also a constant. Therefore, the magnitude on the Richter scale is: \(\boxed{R = -8.16 - \log I_{0}}\)