= 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 satisfies
$$
\operatorname{Re}\log z=2\iff\ln|z|=2\iff\boxed{|z|=e^2.}
$$
The <locus> is \b[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.
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