Seems quite unlikely to me. For a given $n$, the value of $\log a - \log r$ can span a very wide range. I expected it will be impossible to prove $\log a - \log r < n - s_2(n) -2$ for all $a,r$, and it will be impossible to prove $\log a - \log r > n - s_2(n) - 2$ for all $a,r$, because neither one is true. You will need to exclude the specific value $n - s_2(n) - 2$, and you will need to use arguments beyond size considerations (i.e. number theory).