Definition of $x^r$ for $x\in \mathbb{R}$ and $r\in \mathbb{Q^c}$ makes sense if we define it as a limit of $x^{r_n}$ for a sequence ${\{r_n}\} \subset \mathbb{Q}$ for $\lim_{\infty} r_n=r$. And the reasons are that $x^r$ is a continuous function of $r$ and both $\mathbb{Q}$ and \mathbb{Q^c} are dense in $\mathbb{R}$.
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 $n \in \mathbb{N}$ and $c = \dfrac{1}{n}$ for $n \in \mathbb{N}$.
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 $n \in \mathbb{N}$ and $c = \dfrac{1}{n}$ for $n \in \mathbb{N}$ to any $c \in \mathbb{C}$ doesn't seem logical because $\mathbb{C}$ is much "larger" set [I mean it may not be possible uniquely extrapolate to $\mathbb{C}$ from $\mathbb{N} \cup {\{\frac1n}\}$] 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 $c \in \mathbb{N}$ ?
And, Question 2. Is it possible $e^{c \log z} \ne e^{\log z^c}$ for some pair $(z,c)$?