MHB Equality of natural ln function

AI Thread Summary
The discussion centers on the inequality \( (\ln(n))^4 < n^{1/4} \) for \( n > 1 \), which is claimed to hold only for \( n = 1 \) and \( n = 2 \) when \( n \) is a natural number. Participants clarify that the statement is an inequality rather than an equality. The possibility of generalizing this inequality to \( (\ln(n))^b < n^{1/b} \) for \( n > 1 \) is questioned, suggesting that the original inequality may not be valid for all \( n \). The conversation emphasizes the need for careful consideration of the conditions under which such inequalities hold true. The discussion concludes with a focus on the limitations of the proposed generalization.
tmt1
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I have this equality:

$$ (\ln\left({n}\right))^4 < {n}^{\frac{1}{4}} $$ where $ n > 1$

Can I derive a law from this such that

$$ (\ln\left({n}\right))^b < {n}^{\frac{1}{b}} $$ where $n > 1$ ?
 
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Hi tmt. $\ln^4(n)<n^{1/4}$ is only true for $n\in\{1,2\}$, if $n\in\mathbb{N}$.

By the way, it's an inequality.
 
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