Complexity of many constant time steps with occasional logarithmic steps

Question

I have a data structure that can perform a task $T$ in constant time, $O(1)$. However, every $k$th invocation requires $O(\log{n})$, where $k$ is constant.

Is it possible for this task to ever take amortized constant time, or is it impossible because the logarithm will eventually become greater than $k$?

If an upper bound for $n$ is known as $N$, can $k$ be chosen to be less than $\log{N}$?