MHB Natural Log Inequality: True or Misunderstanding?

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The discussion centers on the inequality $(\ln n)^a < n$ for all values of $a$, with participants debating its validity. It is clarified that the inequality is false when considering the limit as $n approaches infinity. An example provided is $\ln^3(e^2)$, which illustrates that the inequality does not hold universally. The conversation emphasizes the importance of context, particularly in relation to limits, and the misunderstanding arises from not specifying the conditions under which the inequality is evaluated. Ultimately, the consensus is that the inequality does not hold true as $n approaches infinity.
tmt1
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I was talking to my professor and she said that $(ln n)^a < n$ for all values of $a$. Is this true or was I misunderstanding?
 
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It's false. Consider $\ln^3\left(e^2\right)$ and note that $e^2\approx7.4$.
 
greg1313 said:
It's false. Consider $\ln^3\left(e^2\right)$ and note that $e^2\approx7.4$.

I forgot an important detail. This is in context of $n$ approaching infinity.
 
You 'forgot' that? You completely denied it when you said "for all n"!
 
This is essentially a question of limits - moved to Pre-Calculus.

tmt, do you think it's true? False? Explain your reasoning.
 

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