Just set it up and solve it. After introducing logorithms, it is just a quadratic equation. It's a mess, but it's not any trickier than the Quadratic Formula.
There are two complications, both are easily resolved.
1) What to do with the solution you don't want and how to identify it. The desired intersection is the one between the means.
2) If the variance matches, the quadratic solution is no good. You'll have to rely on symmetry.