Do the classes $P$ and $NP$ include problems faster then polynomial?
We have $$\mathsf{P} = \bigcup_{k \in \mathbb{N}} DTIME(n^k)$$
And e.g. $\log n \in \mathcal{O}(n)$. Thus, problems which are solvable in deterministic logarithmic time are in $\mathsf{P}$ since they are in $DTIME(n)$.