= Solution
A <branch of the complex logarithm> must be specified to obtain a unique locus. First use the <principal complex logarithm>. Away from the negative real axis, write $z=re^{i\theta}$ with $r>0$ and $-\pi<\theta<\pi$. Then the <complex logarithms> of $z$ and its <complex conjugate> are $\log r+i\theta$ and $\log r-i\theta$. Equating real and imaginary parts gives $\log r=\theta$. Consequently \b[the principal-branch locus is the open segment of a <logarithmic spiral>]
$$
\boxed{z=e^{(1+i)\theta},\qquad -\pi<\theta<\pi.}
$$
The origin is excluded because its <complex logarithm> does not exist. If the principal value is assigned argument $\pi$ on the negative real axis, a point there equals its own <complex conjugate>, so its principal <complex logarithm> would have to equal $i$ times itself. That forces the <complex logarithm> to be zero and cannot hold there; the endpoints remain excluded.
If the printed equation instead means that some values of the multivalued <complex logarithms> satisfy it, write those values as $\log r+it$ and $\log r+iu$, with $e^{it}=z/r$ and $e^{iu}=\overline z/r$. Equality gives $\log r=-u=t$, hence $u=-t$, which is compatible with conjugation. Every real $t$ is allowed. Under this interpretation \b[the locus is the entire <logarithmic spiral> $z=e^{(1+i)t}$, $t\in\mathbb R$]. The PDF does not choose between these branch conventions.
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