Past exam of the mathematical methods I course of the University of Oxford / 2016 / 1

The nonnegative modulus $r=|z|$ must be a zero of the sine function. The restriction $r<10$ permits $r=0,\pi ,2\pi ,3\pi$, since $3\pi <10<4\pi$. Thus the locus is the origin and three concentric circles, centred at the origin with radii $\pi$, $2\pi$ and $3\pi$. A circle of radius zero is just one point.

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.