Addition, multiplication and power are formed in a similar way. One follows from the other by repeating the preceding action several times. But while $a+b=b+a$, and $a \times b=b \times a$, in turn $a^b \neq b^a$ - the power action has no commutativity.
So, I'm curious: is it possible to "flip" $a^b$ into $b^a$ by some action? Like for example $a-b=-(b-a)$ or $\frac{a}{b}=1:\frac{b}{a}$. I understand that subtraction and division are sort of the opposite of addition and multiplication (I apologise for not knowing the terminology). That's why I'm curious: is it even possible?
As I understood, because of the lack of this commutativity, the power has a logarithm and a root (and the root is a variation of the power as $\sqrt[b]{a}=a^{\frac 1 b}$). And with the logarithm you can do just that: $\log_b{a}=\frac{1}{\log_a{b}}$.
So it should be clarified that let a and b are numbers that are not powers of other numbers (like prime numbers just for power) to avoid non-uniqueness of the function of interest. Like here.
Since one number corresponds to exactly one representation by the product of prime multipliers, when we power a number, we are essentially multiplying the power of each prime divisor by the power to which we are do the original number. And we get a new set of prime factors and, as a consequence, a new unique number. Hence the lack of commutativity. And hence the unknowns for me.
What it can mean, how to perform that transformation (if it is possible), and whether I am not mistaken at all? - that's what I'm curious in!