How do I sketch $\operatorname{Re}(e^z)=1$

Question

What is a good method for plotting the graph $\operatorname{Re}(e$$^z$)$=$$1$?

Answer 1

Use that fact that, if $x,y\in\mathbb{R}$,$$\operatorname{Re}(e^{x+yi})=\operatorname{Re}\bigl(e^x\cos(y)+e^x\sin(y)i\bigr)=e^x\cos(y).$$