What does power of logarithm (log) mean?

Question

Here an example of what i am asking

$$\log_2^08=\,?$$

$$\log_2^28=\,?$$

$$\log_2^{-1}8=\frac1{\log_28}=\,?$$

First question is $\log^0 8$ at base $2$

Second question is $\log^2 8$ at base $2$

Also $\log^{-1} 8$ at base $2$ is equal to $1/\log8$ at base $2$?

edit i am asking in analysis of algorithms context. so it is about time complexity

Answer 1

$$\log_2^08=(\log_28)^0=3^0=\,?$$$$\log_2^28=(\log_28)^2=3^2=\,?$$$$\log_2^{-1}8=(\log_28)^{-1}=3^{-1}=\,?$$

Answer 2

It depends. Some times $\log_2^38$ means $(\log_2 8)^3$, some times it means $\log_2(\log_2(\log_2 8))$. Without more context it's impossible to tell.

Answer 3

Simply $$\log_a^bx=(\log_ax)^b.$$

Answer 4

The standard interpretation is

$$\log_a^n b=(\log_a b)^n$$