Prove that $2 ^ {((\log n)^2) }= n \log n$

Question

I've let $y = 2 ^ {((\log n)^2)}$ and arrive at $\log_2 y = (\log n)(\log n)$, but am unsure how to proceed from here. Help will be appreciated thank you!

Answer 1

They are not equal.

$$2^{\log^2(n)}\neq n\cdot\log(n)$$

Substituting $n$ for $1$:

$$\begin{align} & 2^{\log^2(1)}\neq1\cdot\log(1) \\ \implies& 1\neq0 \\ \end{align}$$

Answer 2

Assuming $n$ is a natural number and the logs are $2$-based, your expression is wrong: $$2^{(\log n)^2} =2^{(\log n)(\log n)} = (2^{\log n})^{\log n} =n^{\log n} \neq n\log n.$$