MHB Equality of natural ln function

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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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Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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