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 $c>0$.
2026-03-27 18:34:47.1774636487
What is a good estimate for this log sum?
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Your sum is $\sum_i(\log n-i\log 3)=K\log n-\frac{K(K+1)}{2}\log 3$.