How do I solve the recurrence
$$T(n) = T(n-2) + \log(n)$$
with the condition that $T(n) = O(1)$ for $n \leq 2$?
I started by using an iterative method
$$T(n-2) = T(n - 4) + \log(n-2)$$
then substituting this into the first equation, we find
$$T(n) = T(n - 4) + \log(n-2) + \log(n)$$
Inductively, the next iteration would be
$$T(n) = T(n - 6) + \log(n - 4) + \log(n - 2) + \log(n)$$
and so I saw following pattern
$$T(n) = T(n - 2k) + \sum_{i = 0}^{k - 2} \log(n - 2i)$$
Is this pattern correct, and if so, how should I precede with next step? I am not sure how to transform this sum.