So you want to solve ln(I)/ln(10)= 8.3 so that ln(I)= 8.3 ln(10).

Now, use the "laws of logarithms": [itex]ln(I)= ln(10^{83})[/itex]

If the Richter scale is doubled, that is 16.6 rather than 8.3, then you need to solve ln(I)/ln(10)= 16.6 so [itex]ln(I)= 16.6 ln(10)= ln(10^{16.6}[/itex]. Once you have found this I, divide by the previous one to find the factor.