I have the following non-linear equation:
$e^{\beta_0+\beta_1x+\beta_2*y}+C = e^{\mu_0+\mu_1x+\mu_2*y}$.
$\beta_i$ is known, but $\mu_0$, $\mu_1$ and $\mu_2$ are parameters we want to estimate. x and y are two binary variables, x={0,1} and y={0,1}. For the different combinations of x and y we get the following 4 equations:
1) x=0, y=0: $e^{\beta_0}+C = e^{\mu_0}$.
2) x=1, y=0: $e^{\beta_0+\beta_1}+C = e^{\mu_0+\mu_1}$.
3) x=0, y=1: $e^{\beta_0+\beta_2}+C = e^{\mu_0+\mu_2}$.
4) x=1, y=1: $e^{\beta_0+\beta_1+\beta_2}+C = e^{\mu_0+\mu_1+\mu_2}$.
C is a constant we can manually adjust. Does this system of equations have a solution?
Hint: Start with your first equation and use it to eliminate $\mathrm{e}^{\mu_0}$ from the other equations.
Then use the second equation to eliminate $\mathrm{e}^{\mu_1}$ from the other equations.
At the end you will get one equation with beta only. You will need to check if this equation can be true for a particular choice of $C$. If that is not possible you will need to use non-linear regressions to get parameters $\mu_j$.
Main idea of non-linear regression (here non-linear least squares): Define a new function $$F(\mu_0,\mu_1,\mu_2,C)=(e^{\beta_0}+C-e^{\mu_0})^2+(e^{\beta_0+\beta_1}+C - e^{\mu_0+\mu_1})^2+(e^{\beta_0+\beta_2}+C - e^{\mu_0+\mu_2})^2+(e^{\beta_0+\beta_1+\beta_2}+C - e^{\mu_0+\mu_1+\mu_2})^2.$$
The goal is to minimize this function (this is a non-linear least squares solution). So you need to find the gradient of $F$ and set it to zero. The resulting system of equations can be solved by using for example Newton-Raphson procedure. Note that this procedure is quite inaccurate for only four data points. If you have more datapoints you can try to get a better fit.