# What is a good estimate for this log sum?

Given $$n\gg0$$ what is a good estimate for $$\sum_{i=1}^K\log\Big(\frac{n}{3^i}\Big)?$$

I am particularly interested in case of $$K=O(1)$$ and $$K=O((\log n)^c)$$ at a fixed $$c>0$$.

Your sum is $$\sum_i(\log n-i\log 3)=K\log n-\frac{K(K+1)}{2}\log 3$$.