Is $\log^22^k = (\log2^k)^2$?

Question

Does the square of $\log^22^k$ include whole $2^k$? Is $\log^22^k = (\log2^k)^2 = \log2^k \cdot \log2^k = k^2$?

The base of my $\log$ is $2$.

Answer 1

I think $ \log^22^k$ is a very confusing notation.

I'd rather go with $$ (\log2^k)^2 = (\log2^k)\times(\log2^k) = k^2(\log2)^2 $$

But again its down to preference.

If we're considering base 2, yes, the answer simplifies to $k^2$

Answer 2

$$\log_2 2^k = k$$

Squaring both sides:

$$\log_2^2 2^k = k^2$$

The expression $(\log_2^2 2)^k$ is not a useful expression as it simply evaluates to one, so the above interpretation is almost certainly what one would mean when writing $\log_2^2 2^k$

Answer 3

There's a slight flaw in your working out. $$ \log2^k \cdot \log2^k = k^2 (\log2)^2$$ Since we have no reason to suppose that $(\log2)^2=1$, I'd have to say that your last step is causing the error.

Answer 4

$\log^2 x$ notation is a very annoying one. People use it for both $\log \log x$ and $(\log x)^2$. Here's my theory on how this notational nightmare came about.

In general, with a function $f$, we tend to expect $f^2 (x)$ to be $f(f(x))$ (composition) while $(f(x))^2$ means exactly that - the value of $f(x)$ squared.

However, notation of $\sin x, \cos x, \tan x$ developed where $\sin^2 x, \cos^2 x, \tan^2 x,$ meant $(\sin x)^2, (\cos x)^2, (\tan x)^2$. In this case, the error is not so egregious since it is extremely uncommon to see $\sin \sin x$, since it doesn't really make sense as a concept. That is, $\sin x$ takes in an angle and gives you a ratio, so it doesn't usually make sense to then take the $\sin$ of a ratio when its argument should be an angle.

Then, people used $f^2(x)$ for functions like $\log$ to mean $(f(x))^2$ since they were used to doing it for trigonometric functions and failed to see the nuance above of why it works out in that case.

EDIT: Okay, now I'm confused. Is OP asking if $\log^2 2^k = \log^2 (2^k)$ or if $\log^2 2^k = (\log^2 2)^k$ or is he asking about the exponent of $2$ over the $\log$? His first statement implies the first but his second statement seems to imply the latter...