I'm stuck with an exercise. Let $t$ be real, $t > 1$, then:
$te^{t-z}=1$ has a unique solution $z_0$ in the unit disk. Also, $z_0$ is real.
My solution is $z_0=\log(t)+t$ obviously, but if $t > 1$, then this is real, and strictly greater than $1$, hence not in the unit disk? So I do not get behind the existence.
(Logarithm has period $2\pi i$ in the complex plane, and only one of the points $z_0, z_0+2\pi i, z_0-2 \pi i, ...$ can be in the unit circle, since its diameter is $2$, so the uniqueness is no problem.)