What does $\log^{2}{x}$ mean? What is it used for and why  doesn't it equal $\log{x^2}$?
 A: It is convention to write 
$$\log ^2 x$$
in place of 
$$(\log x)^2=\log x \times \log x$$
just in the same manner as we write $\sin^2 x = (\sin x)^2$ or any other "named" function's square. In general, you will see $(f(x))^n$ for the $n$th power of a function, and  $f^n (x)$ in place of the $n$ times composition of $f$, i.e. $f^3(x)=f \circ f \circ f(x)$.  However, for the $\log$ and trigonometric functions, we break this convention.
Note that $\log(x^2)=2 \log x \neq \log x \cdot \log x$
A: Already "log" is ambiguous, implying respectively base 10, e, or 2 in elementary applied mathematics, general mathematics, and information theory or theoretical computer science. In the second case, particularly in number theory and theoretical statistics, the iterated logarithm arises naturally, and some authors mean $\log \log x$ by $\log^2 x$. The squared logarithm isn't seen much outside school calculus textbooks. The reverse is true for trigonometric functions: the squared functions are ubiquitous, while the iterated functions are mostly confined to examples and exercises for students. The notation $\sin^2 x$ is illogical (Gauss, in particular, complained about it), but so convenient and established by tradition that we are probably stuck with it.  
