Natural Log Inequality: True or Misunderstanding?

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SUMMARY

The inequality $(\ln n)^a < n$ is false for all values of $a$ when considering the limit as $n$ approaches infinity. A counterexample is provided with $\ln^3(e^2)$, where $e^2 \approx 7.4$. The discussion emphasizes that the misunderstanding arises from not specifying the context of $n$ approaching infinity, which is crucial for evaluating the inequality correctly. This topic is fundamentally linked to the concept of limits in calculus.

PREREQUISITES
  • Understanding of logarithmic functions and their properties
  • Familiarity with limits in calculus
  • Basic knowledge of exponential functions
  • Concept of asymptotic behavior in mathematical analysis
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  • Study the properties of logarithmic functions in detail
  • Learn about limits and their applications in calculus
  • Explore asymptotic analysis and its significance in mathematical proofs
  • Investigate counterexamples in mathematical inequalities
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Students studying calculus, educators teaching logarithmic functions, and anyone interested in understanding mathematical inequalities and limits.

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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