I have tried solving by using Taylor's series expansion for $e^x$ and approximating till the third term of the expansion (i.e. term with power 2). This gave me a quadratic equation which was easy to solve. But I am using the solution of this equation as part of a bigger numerical simulation using Finite Difference method. And the small error due to approximation in the expansion for $e^x$ accumulated over many iterations and gave erroneous results in the end.
So, I need some other way of solving this equation that does not involve approximation.