[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 ;) ).