I hope to not seem to much silly but:
are the same or not?
The usual notation for "log x times log x" is log^2 x (similarly to sin^2 x, means "sin x times sin x").
The usual notation for " the log of (x times x)" is log x^2.
In your examples the () are not really needed unless "x" is a more complicated expression, like log^2(x+y) or log (x+y)^2.
BTW it's not a silly question - if you don't know what any mathematical notation means, you need to find out, otherwise you can't use it properly!
What is log^2(x)?
Did you mean this: [tex]\log_2(x)[/tex]? In that case they may not.
[tex]\log_2(x)[/tex] is the logarithm of x to the base 2.
[tex](\log(x))^2[/tex] is the square of the logarithm of x to some base, usually taken to be 10.
[tex](\log(x^2))[/tex] is the logarithm of x^2. Again, the base may be 10.
I've never come across that notation before!
I'm with neutrino: I've never seen log(x)2. Either log2(x) or (log(x))2 means (log(x))(log(x)). log(x2) means, of course log((x)(x)) (which is equal to 2log(x)). Of course, since people do not always use parentheses with logarithm (or trig functions), you might see log x2 which I would interpret as log(x2)= 2log(x).
Well, at least nobody's suggested that log^2 x meant log(log x) - and it doesn't, mean that, of course.
I can't think of a reason why you would ever want to write (log x)^2 except in a calculus exercise like "integrate the function (log x)/x".
But if I did see it, I would have assumed it meant the same as sin^2 x etc, unless the context implied something different.
Err...it's the log2(x) that I haven't come across. It has always been (log(x))2 for me.
Separate names with a comma.