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I would love some help with this question please:

Sketch the image under the function $w = \log z$ of the following set $\{z : Im z > 0\}$

What I have done so far is as follows:

$R= \{x+iy: y>0, x \in {\bf R}\}$

$R= \{re^{iθ}: 0<θ\le \pi,r>0\}$

$f(R)= \{~\log re^{iθ}: 0<θ\le \pi,r>0,\}$

$=\{\log r + iθ: 0<θ\le \pi,r>0\}$

I am unsure if this is correct and if it correct would the sketch be all the area above the x axis from $0$ to $\pi$?

Also would the sketch exclude the origin since $\log(0)$ is not defined and would it also exclude the line of the x-axis since $θ>0$?

I hope that my question is clear and I would really love any help on this.

Many thanks.

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