I am looking to find a way to a way to solve for a combination of optical filters that match a transmittance dataset that I have. I need to solve for the exponents x,y,z...n (they can be zero) as well as quantify how close I got to the target transmittance curve across it's domain (300-700nm). Here the exponential variables represent the number of times each filter is chosen and there is a .96 (loss) in the gap between filters.
$$T_{total} = T_1^{x}*T_2^{y}*\cdots *T_{46}^{n}(0.96^{[(x+y+⋯+n) -1]}) $$
I was going to do $X= A^{-1}*B$ in matlab but I think that only works for forms like $T = aX + bY \cdots nZ$
I am really unfamiliar with this type of math and don't know where to start.
You can transform the problem by taking a logarithm, which will give you something like:
$\begin{eqnarray}\log T_{total} & = & x \log T_1 + y \log T_2 + \ldots + (x + y + \ldots - 1) \log 0.96 \\ \log T_{total} + \log 0.96 & = & x (\log T_1 + \log 0.96) + y (\log T_2 + \log 0.96) + \ldots \end{eqnarray}$
And then you can just do a linear fit on the values of $\log T_{total} + \log 0.96$ against $(\log T_1 + \log 0.96, \log T_2 + \log 0.96, \ldots)$.