What is the difference between $\log_2^2 x$ and $(\log_2 x)^2$? And what methods should be used for solving equations with the first case. For second I use u-substitution method in equations that have logarithm raised by different powers.
For instance,
$\log_2^2 x - \log_2 x = 2$
(We asume that $\log_2^2 x = \log_2 (log_2 x)$)
It would be nice, if you could also explain, how to solve this one.
Usually $\log^2$ is the same as $(\log)^2$.
This is because the $log$ function is a function, and you may denote it as $f$. Hence
$$f^2 = (f)^2$$
The other expression you wrote, namely $\log(\log)$ is not a square; this is a composition, namely (in terms of $f$) it's a $$f\circ f$$
And of course we have that in general
$$f\circ f \neq f^2$$
I have never seen the notation $\log^2$ used to mean $\log(\log)$. It would be messy and ill defined.