[gms7777]

I agree that log^2(x) is more commonly used to refer to log(x)log(x) than log(log(x)), though I've seen it both ways (I think its somewhat dependent on the conventions of the field). That said, the notation is both ambiguous and non-sensical (you're not squaring the operator...), and at least in my opinion, should be avoided in favor of the less ambiguous (log(x))^k

Gauss actually had some choice words about the notationally similar sin^2(x): "Sin2 φ is odious to me, even though Laplace made use of it; should it be feared that sin2 φ might become ambiguous, which would perhaps never occur, or at most very rarely when speaking of sin(φ2), well then, let us write (sin φ)2, but not sin2 φ, which by analogy should signify sin (sin φ)"

[aweyl]

Using log^2(x) to denote log(log(x)) is far from nonsensical. It makes sense when keeping in mind the fact that function composition is an operation on functions much like addition and multiplication. For example, we can write f(f(x)) = (f∘f)(x) = f^2(x) f(f(f(x))) = (f∘(f∘f))(x) = (f∘f∘f)(x) an so on. We can see that when composing a function with itself many times it is much easier to write f^n(x) than the alternative.

Using log^2(x) to mean (log(x))^2 is a shortcut that typesetters took in order to save time and is not as widely used in mathematics.

[gms7777]

I agree with you. I only meant to refer to the (log(x))^2 case as nonsensical, though I think my wording was ambiguous. I understand its a hort-cut but it doesn't make mathematical sense as a notation. I think it makes more in the log(log(x)) case, for the same reasons you've pointed out. Though, at least in my experience in my field (theoretical CS), I see it used as (log(x))^2 far more often than log(log(x)).

[pjakma]

Yes, log^2(x) seems almost always to be used to mean [log(x)]^2 in the routing theory papers I tend to read.

I agree with you though that using log^2(x) for [log(x)]^2 is horridly ambiguous notation that should be avoided! (Which reminds me, I should go and check my thesis for this ;) ).