The magnitude $R$, measured on the Richter scale, of an earthquake of intensity I is defined as $R=\log \frac{1}{I_{0}}$, where $I_{0}$ is a minimum intensity used for comparison. If the intensity of an earthquake was $10^{8,16} \cdot I_{0}$, what was the magnitude on the Richter scale?

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: \(\boxed{R = -8.16 - \log I_{0}}\)