The logarithm of a sum is not something you can simplify. The identity only works the other way.
With this problem, I'd probably go for a numerical solution using Newton-Raphson. Of course, this assumes you know a, b, and c, and that a and b are both positive. Is that the case?
Right, but you can always convert an equation-solving problem into a root-finding problem by throwing everything on to one side of the equation thus:
Then you use Newton-Raphson on
If you have convergence problems, and your solutions are oscillating wildly, then you probably don't have a solution for that particular combination of a, b, and c. That's a little warning sign you can look for.