I need to find a real solution for $\theta$ in the following complex equation:
$e^{(k+i)\theta} + e^{(k-i)\theta} = e^{k\theta} + c$
Here $k$ and $c$ are positive real numbers.
The problem is related to the problem of finding a log spiral given by $r = r_0 e^{(k+i)\theta} + z$ with known $r_0$ and $k$, which passes through two points $z_0$ and $z_1$. In the equation, c = $\dfrac{d^2}{r_0^2}$ where $d$ is the distance between $z_0$ and $z_1$.
An approximate solution (with given error bound) can also be useful.
A more general question about such spirals is here