Exponent log inequalites

Question

I'm stuck in trying to figure out why this inequality is true: $$ n^{\log^4 n} \leq 2^{\log^5 n} ~~~~~~ \text{for} ~n>2$$ (Here $\log$ denotes the base $2$ logarithm). I'm sure there's some simple little algebraic trick but I'm just not seeing it. Can someone help me out?

Answer 1

Just take log both sides it becomes $\log^5 n $

Answer 2

$$n^{\log^4 n} \leq 2^{\log^5 n} $$ $$2^{\log(n)\log^4 n} \leq 2^{\log^5 n} $$ $$2^{\log^5 n} \leq 2^{\log^5 n} $$ They are equals.. or $$n^{\log^4 n} \leq 2^{\log^5 n}=2^{\log(n)\log^4 n}=(2^{\log(n)})^{\log^4 n}=n^{\log^4 n}$$