Natural Log Inequality: True or Misunderstanding?

[tmt1]

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?

 

[Greg]

It's false. Consider $\ln^3\left(e^2\right)$ and note that $e^2\approx7.4$.

 

[tmt1]

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.

 

[HallsofIvy]

You 'forgot' that? You completely denied it when you said "for all n"!

 

[Greg]

This is essentially a question of limits - moved to Pre-Calculus.

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