Error in approximation to log(223)/log(3) .... senior moment?

This is probably a silly question, but I am really stuck. A possible senior moment, is my only excuse.

Here is an approximation:
##log(223)/log(3) \approx 10818288 / 2198026 ##

So we have:
##log(223)/log(3) - 10818288 / 2198026 = 0.0399292##
which is OK but not great -- the error shows up right at the second decimal.

But when we do this:
##10818288 \times log(223) - 2198026 \times log(3)## it gives us -0.000984652, which looks way better.

I would expect the error between two large numbers to be larger than when the same thing is recast as a difference between two small numbers. Again, it's probably a silly thing that I'm missing, but I haven't been able to find it.