Simon,
There is no logarithm rule that allows you to solve for x explicitly. However, from differentiating we can see that if a*b and c*d are either both positive or both negative, then y(x) is monotone. This means that y(x) is a one-to-one function and has a well defined inverse. In this case, the function fzero can be used to determine the an x value from a y value. Here's how.
% Determine constants from a fitting routine. Dummy data here.
a = 1;
b = 2;
c = 3;
d = 4;
% Create an anonymous function for representing the fit.
f = @(x) a*exp(b*x) + c*exp(d*x);
% For a y value of interest, find x.
y_given = f(1); % This example y value corresponds to x = 1.
x_calculated = fzero(@(x) f(x) - y_given, 0);
Enjoy,
Jonathan
