In my research I have come across the equation
$$\prod_i^n \left( \frac{a_i}{x} \right)^\frac{b_i}{x} = \prod_i^n (1-d_i)^{(1-b_i)c}$$
Is it possible to obtain $x$ from this analytically, or do I have to search numerically using e.g. a root-finding algorithm?
The solution I seek for $x$ must be strictly positive (in fact all variables are positive), and $b \in [0,1]$.
I doubt it is possible (EDIT: I was wrong), but it can be simplified quite a bit. Do a change of variables $y=1/x$, and take the logarithm of both sides: $$\sum_i (b_iy)\log(a_iy) = \left(\sum_i b_i\right) y\log y + \left(\sum_i b_i\log a_i\right) y = \log \prod_i^n (1-d_i)^{(1-b_i)c}.$$ So you have a problem of the form $\alpha y\log y + \beta y = \gamma$. That's not going to be difficult to solve numerically. There might be a way to express the solution in terms of the Lambert W function, but in most computational environments you're not likely to find that implemented. (EDIT: See Robert's formula below; I got the same result from Wolfram Alpha.)