I have the following question: Sketch the image under the map $w = \log z$ of $|z| = 1$.
I know that its a unit circle and by the definition $\log z=\log |z|+i \operatorname{Arg} z, \quad z \neq 0$ . How do I map a circle with log function? The only think I know is that the circle go from $0$ to $2\pi$ so the Imaginary part of w should be $-\pi$ to $\pi$ by the definition since it is $i\operatorname{Arg}z$. But how do I compute the real part? The definition says it should be $\log|z|$ but how do I relate a $|z| = 1 $ to $\log|z|$?
So the question is: how do I relate a $|z| = 1 $ to $\log|z|$. ? The result is $0$ from the solution Manuel so combining the real and imaginary I get a line at $x=0$ where the $iy$ values from $-\pi$ to $\pi$ but I don't understand the real part. Any help is appreciated.