Vtyjkyuk

Definition of $x^r$ for $xin mathbbR$ and $rin mathbbQ^c$ makes sense if we define it as a limit of $x^r_n$ for a sequence $r_n subset mathbbQ$ for $lim_infty r_n=r$. And the reasons are that $x^r$ is a continuous function of $r$ and both $mathbbQ$ and mathbbQ^c are dense in $mathbbR$.

Churchill's book defines $z^c = e^c log z$ because

[it] provides a consistent definition of $z^c$ in the sense that it is
already known to be valid when $c =n$ for $n in mathbbN$ and $c =
dfrac1n$ for $n in mathbbN$.

Also the book defines $log z$ based on $z = e^log z$. So, my Question 1. is that isn't it more reasonable to to define $z^c = e^log z^c$ since

1- "Defining" a number (or a set of numbers) makes sense only if it indicates the value of it and we can't define whatever we want;

2- Extending from $c =n$ for $n in mathbbN$ and $c = dfrac1n$ for $n in mathbbN$ to any $c in mathbbC$ doesn't seem logical because $mathbbC$ is much "larger" set [I mean it may not be possible uniquely extrapolate to $mathbbC$ from $mathbbN cup frac1n$] and it may result in different extensions.

3- Though may(?) $e^log z^c=e^c log z$ but $log z^c ne c log z$ in general even for $c in mathbbN$ ?

And, Question 2. Is it possible $e^c log z ne e^log z^c$ for some pair $(z,c)$?