Why a Precision problem for Floor of quotient of Logs?

Question

This line returns a Precision problem:

Floor[Log[9]/Log[3]]

This one works fine:

Floor[Log[9]/Log[3]//N]

What I don't understand is why one works while the other doesn't because I assume that internally Mathematica would numerically evaluate the argument of Floor first anyway, so the two lines should be logically equivalent, the 2nd one just redundant. But clearly I was wrong. What did I misunderstand about how Floor works?

Answer 1

While the comments already contain a complete answer by Daniel Lichtblau, this behavior is also documented on ref/Floor, see the first example under 'Possible Issues':

Floor does not automatically resolve the value:

Floor[π^2 + 2 π + 1 - (π + 1)^2]

(* Floor::meprec: Internal precision limit
$MaxExtraPrecision = 50.` reached while evaluating Floor[1+2π+π^2-(1+π)^2]. *)

(* Floor[1 + 2 π + π^2 - (1 + π)^2] *)

Even though 1 + 2 π + π^2 - (1 + π)^2 is in fact the exact integer 0, it is a "hidden zero" which cannot be unmasked using purely numerical methods.

Unlike other functions like FullSimplify and PossibleZeroQ which may use symbolic algorithms, Floor is limited to numerical methods, see also this documentation note

For exact numeric quantities, Floor internally uses numerical approximations to establish its result.

and the numerical approximation will never yield an exact zero, no matter how much extra precision is allowed, e.g.

Block[{$MaxExtraPrecision = 1000}, N[π^2 + 2 π + 1 - (π + 1)^2, 50]]

Similarly, Log[9]/Log[3] is a "hidden integer". Numerical approximation of the exact quantity Log[9]/Log[3] - 2 is difficult, while numerical approximation of the machine precision number N[Log[9]/Log[3]] - 2 is not.