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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!

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    $\begingroup$ try $n=1$ first. You obtain $2^0=1\cdot0$ i.e. $1=0$. From here you could proceed to prove anything, by a contradiction. $\endgroup$ – Mirko Apr 26 '18 at 2:40
  • $\begingroup$ Is that log base 2 or some other base. $\endgroup$ – fleablood Apr 26 '18 at 5:46
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They are not equal.

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

Substituting $n$ for $1$:

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

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

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