This doesn't contradict what I said. I said that the odds of a function needing more than 3n digits was 4^-n. For single precision, that means you would expect almost all functions to be OK with 3n bits (1 in 2^46 wouldn't be).
In my experience, low order bits of transcendental functions make terrible rng sources (even discounting their runtime). Your example of the logarithm is great -- for large inputs, you need large changes to flip even low order bits.