# Exponential Equilibrium Equation

1. Dec 17, 2008

### ArbyFisher

Solve for real-valued x, e-aX + e-bX = 1, where a and b are arbitrary known constants > 0.

For example, e-48.12/50 + e-48.12/100 ~ 1.00

In this case X = 48.12 (to two decimals), a = 1/50 and b = 1/100.
For any specific values of a and b, a computational solution can easily be determined, but a general algebraic solution is desired.

The problem can be variously reformulated, i.e., Y = eX, Yc + Yd = 1, c and d < 0

... alas to no apparent avail. The equation sometimes seems like it should be a queuing theory probability or perhaps some geometric shape or I don't know anymore (obviously) ... thanks in any case.

2. Dec 17, 2008