Quick problem; can't find other solution

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SUMMARY

The discussion centers on solving the equation a = e^-t t^x, which is expected to have two real solutions between 0 and e^-x x^x. One solution has been identified as t = -x ProductLog[-(a^((1/x))/x)], utilizing the Lambert-W function, also known as the Product Log. The challenge lies in determining the second solution, which is related to the multivalued nature of the Lambert-W function and its branches for real-valued arguments.

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ayae
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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.
 
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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.
 

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