MHB Equality of natural ln function

Thread starter tmt1
Start date Jul 16, 2016
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Function Ln Natural

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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.

Jul 16, 2016

tmt1

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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Jul 16, 2016

Greg

Gold Member

MHB

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.

Last edited: Jul 16, 2016

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For the sake of argument, lets assume this is, in fact, a right triangle with a rectangle inscribed in it. Here is what looks like a very elegant solution: Triangles A and B are similar a/4=2/b ab=8 But what is happening in that second line? Is there a property of similar triangles I'm forgetting?

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