as part of proving a question in homework I came across the following logarithmic inequality: $$ \log ^4n<n^{1-\epsilon}$$ I need to find a value of epsilon that the inequality will continue to be true. I checked with wolfy and the value is around 0.3 (can be less, but not greater).
I thought the best way will be to use L'Hopital's Rule on this expression: $$ \underset {n\rightarrow \infty}{lim}\frac {\log n}{n^\frac{1-\epsilon}{4}}=0 $$ and then if that's true than there'll be an $n_0\in \mathbb{R}$ that from it and on the expression: $\frac {\log n}{n^\frac{1-\epsilon}{4}}<1$ will be true but it doe not give me any information on epsilon (except that it needs to be less than 1).
Hint: instead of worrying about limits at infinity, find where the function $$f(x) = \frac{\log~x}{x^\alpha}$$ takes on its maximum value.