In practice the low-order bits of transcendental functions are pretty much indistinguishable from random for _most_ inputs. You do have to be very careful close to especially nice values though, because this falls apart catastrophically there (e.g. logarithms close to 1)--it should be pretty obvious from the Taylor expansions why this happens. The good news is that when this does happen, almost by construction you usually don't end up very close to an exact halfway point for rounding.
> In practice the low-order bits of transcendental functions are pretty much indistinguishable from random for _most_ inputs.
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.