Solving inequality having log I am struggling to solve this inequality involving logarithm. How to

Chaz Blair

Chaz Blair

Answered question

2022-05-21

Solving inequality having log
I am struggling to solve this inequality involving logarithm. How to find out values of n for which below inequality holds good:
log 2 n n > 1 8

Answer & Explanation

vishaex0

vishaex0

Beginner2022-05-22Added 6 answers

Rearrange this to state that
log n > n 8
since clearly n > 0. Exponentiate and find that
n > 2 n / 8
or alternatively,
n 8 > 2 n
This can be solved exactly using the Lambert W function, or done numerically to find that the upper bound is about 43.56 as here or here.
Wayne Steele

Wayne Steele

Beginner2022-05-23Added 1 answers

I proved a more general result as an answer to a similar question:
If n and k are integers and k 2 and n k 2 + 1, then 2 n > n k
For k = 8 his gives 65 which is not too far from the best bound of 43.56.

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