So what I want to have is image of $\log(z)$ for $e^{-\pi/4} < |z| < e^{\pi/4}$.
So from what I have been able to figure out for the image is that the new magnitude of circle will be $\log(|z|)$ while the argument of the number will stay same. However the log of 1 is zero, I'm not able to imagine where the circle with radius of magnitude $e^{-\pi/4}$ which is less than 1.
Thank You Very Much.
Well, it's quite easy to write $z$ down given $e^{-\pi/4}<|z|<e^{\pi/4}$; we can just write
$$z=re^{i\theta}$$
with $e^{-\pi/4}<r<e^{\pi/4}$ a real number and $\theta\in[0,2\pi)$. Now what happens when we throw this into $\log$? Well,
$$\log(z)=\log(re^{i\theta})=\log(r)+i\theta$$
Can you take it from here?