 Solution

For any value of the complex logarithm, $\log z=\ln |z|+i\arg z$ up to adding an integer multiple of $2\pi i$. Its real part therefore satisfiesThe locus is the circle centred at the origin with radius $e^2$. Different logarithm values change only the imaginary part. If a particular analytic branch of the complex logarithm with a cut is imposed, restrict the circle to that branch's domain; the modulus equation itself is branch independent.