If an inequality has solutions or not

Question

I am trying an problem in analysis but the solution depends upon whether the following inequality is solvable or not

$n \geq (\log n)^{3n} $ .

I have no idea on how to find solutions (if any) of this inequality.

Any help will be appreciated.

Answer 1

$$(\log n)^{3n}$$ is a fast growing function and the shortest way to solve the inequation is probably to try the integers systematically.

$$0,0.11\cdots,2.33\cdots,50.38\cdots$$ leaves little doubt.

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If the question is understood as $$n>\log_{10}n$$ the sequence is, starting from $10$,

$$1,3.37\cdots,15.53\cdots$$