Question: > 
> Is $\log{N}^2$ read as $(\log{N})^2$ or $\log{(N^2)}$? 
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> 
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How can I write $(\log{N})^2$ without using parenthesis?

Answer: 
$\log{N^2}=\log(N^2)$.

$(\log{N})^2=\log^2N$.
In general you can write $f^2(x)=(f(x))^2$ but $f(x^2)$ is not $f^2(x)$ in general.

Here $\log N^2=2\log N=\log(N^2)\neq (\log N)^2=(\log N)\cdot (\log N)$, in general (Obviously equality holds for $N=1$ and $N=e^2$ in base $e$)