$(\log^2(n))$ and $(\log(n)^2)$, are both same?

Question

I have two functions, $\log^2 (n)$ and $(\log(n))^2$, is both the functions are same?

Can anyone help me on this?

Edit 1: Is $\log(log( (n))$ is same as $\log^2 (n)$ ?

Sometimes when a function is written as $f^2(x)$ this is interpreted as $f(f(x))$, I think it depends on the context — gd1035, Jul 03 '18 at 17:33
@gd1035 I'm used to that kind of interpretation when there's no "multiplication" in the space of functions. — Gonzalo Benavides, Jul 03 '18 at 17:34
Regarding your edit, no, they are not the same. As an example, let $n=10$. Then assuming base $10$, we have $$\log\log n=\log\log10=\log1=0$$ whereas $$\log^2n=\log^210=1^2=1\neq0.$$ — ə̷̶̸͇̘̜́̍͗̂̄︣͟, Jul 03 '18 at 18:40

score 2 · Answer 1 · answered Jul 03 '18 at 18:23

I must say that for some functions, like logarithm or trigonometric functions, it is traditional to use $f^2(x)$ to mean $f(x)^2$.

But in any other context where you have a function (or any map from a set to itself), $f^2(x)$ means $f(f(x))$, which is not the same that $f(x)^2$.

So the answer is: $\log^2(x)=\log(x)^2$, but $f^2(x)=f(f(x))\ne f(x)^2$ if $f$ is any "general" function. Quite confusing, isn't it?

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