Are there any problems whose best known algorithm has run time $O\left(\frac{f(n)}{\log n}\right)$
The usual answer to "what could cause you to divide by a log?" is a combination of two things:


a model of computation in which constant time arithmetic operations on word-sized integers are allowed, but in which you want to be conservative about how long the words are, so you assume $O(\log n)$ bits per word (because any fewer than that and you couldn't even address all of memory, and also because algorithms that use table lookups would take too much time to set up the tables if the words were longer), and
an algorithm that compresses the data by packing bits into words and then operates on the words.


I think there are many examples, but the classic example is the Four Russians Algorithm
for longest common subsequences etc. It actually ends up being $O(n^2/\log^2 n)$, because it uses the bit-packing idea but then saves a second log factor using another idea: replacing blocks of $O(\log^2 n)$ bit operations by a single table lookup.