Can anyone prove:
$ ⌊log_2N⌋ <=log_{3/2}N$
This question comes from the Binary Tree height calculation. If the number of element in a binary tree is N, the height of the tree is the floor of $ log_2N $ which is smaller than $log_{3/2}N$.
video link: https://www.youtube.com/watch?v=La-xYKoKwjY&t=61s (time: 19:11)
By definition of the greatest integer (floor) function: $$\;⌊\log_2N⌋ \le \log_2N \,$$
By monotonicity of the $\,\ln\,$ function, and given that $\,N\,$ is a positive integer, so $\,\ln N \ge 0\,$: $$\;\ln 2 \gt \ln \cfrac{3}{2} \gt \ln 1 = 0 \;\;\;\implies\;\;\;\log_2N = \cfrac{\ln N}{\ln 2}\lt \cfrac{\ln N}{\ln 3/2}=\log_{3/2}N\,$$
