Therefore where is the increasingly well known Lambert W-function (eg. see Lambert W-Function -- from Wolfram MathWorld. You can search MHF for various other questions where it's been used.)
I'm not quite sure how to tackle this equation:
How do I solve for x? All I know is there is one real value of x, and that is ~.567. Is there a tidy expression for x, and if not, is there a name for this occurrence, or even this type of eqn?