# Definition of the complex $\log^z z$ to know the solutions of $\Re \log^x x=\Im \log^x x$ when $x$ runs over the reals between $0$ and $1$

I would like to know a draft to refresh how to define the complex function $$f(z)=\log^z z,\tag{1}$$ here thus $z\in\mathbb{C}$. Is required a definition of our logarithm here in a way that my purpose explained in next paragraphs has mathematical meaning.

My question arises when I was playing with Wolfram Alpha online calculator with the code

plot log^x x, from x=0 to 1

The exercise that I would like to know in my home is to calculate the corresponding real and imaginary parts $\Re \log^x x$ and $\Im \log^x x$, over the real ray $x>0$, with the purpose to know how many solutions has the equation $$\Re \log^x x=\Im \log^x x\tag{2}$$ over the real segment $0<x\leq 1$ (it is easy to see that the real $x=1$ is a solution, but I would like to know/prove all solutions over reals between $0$ and $1$).

Question. In previous context, can you refresh to me how to define the complex function $$f(z)=\log^z z?$$ Only is required hints, and the definition of previous complex $\log^z z$ is according my purpose to know how many solutions has the equation $\Re \log^x x=\Im \log^x x$ for reals $0<x\leq 1$ . Many thanks.

Thus I need hints or details to get a definition/expresion for $f(z)$, and from here I can to calculate its imaginary and real parts.

• For complex $w$ and a choice $\log$ of branch of logarithm, we can define $w^{\zeta} := \exp(\zeta \log w)$. – Travis Apr 7 '18 at 10:45
• That is, I need a good definition of the branch of the complex logarithmm @Travis – user243301 Apr 7 '18 at 10:46
• See en.wikipedia.org/wiki/… , but you've already implicitly chosen one just by writing down the expression $\log z$ in the first place. – Travis Apr 7 '18 at 10:58
• Many thanks really I need a good refresh of complex analysis @Travis – user243301 Apr 7 '18 at 11:09