Discussion
The set \(\{z \in \mathbf{C} : |e^{z}| = 2019\}\) represents :
Answer: A
The set is the line \(x = \log(2019)\).
For \(z = x + iy\), \(e^{z} = e^{x}e^{iy}\) and \(|e^{iy}| = 1\), so \(|e^{z}| = e^{x}\). The condition \(e^{x} = 2019\) gives \(x = \log 2019\) with \(y\) arbitrary: a vertical line in the plane.
The modulus does not depend on \(y\) at all, so \(y = 2019\) is wrong, and the set is neither a circle nor countable (it contains a point for every real \(y\)). A circle would come from \(|z| = \text{const}\), not \(|e^{z}| = \text{const}\).
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