Quick problem; can't find other solution

[ayae]

I'll make this quick guys:
a = e^-t t^x
This equation should have 2 solutions real between 0 and e^-x x^x
I've found one solution to the equation:
t = -x ProductLog[-(a^((1/x))/x)]
But I can't find the other :(. Please can you give me a hand finding the other solution.

 

[Mute]

The "Product Log", or Lambert-W function, is multivalued, so you have to choose a branch. For real valued arguments there are two branches; I would suppose this is your what your second solution is.