This question uses the following definition of a complete binary tree†:
A binary tree $T$ with $N$ levels is complete if all levels except possibly the last are completely full, and the last level has all its nodes to the left side.
The following is an excerpt from Algorithms:
It ($\log N$) is also the depth of a complete binary tree with $N$ nodes. (More precisely: $⌊\log N⌋$.)
Why is the above excerpt true?
† Originally defined here