Determine all possible real x that satisfy this equation:
(1/x)^{x} = 4^{x+1/16}
I'm not sure whether there is another actual solution but the LHS (1/x)^x is defined for negative rational numbers with an odd denominator.
The RHS is always positive, so the rational solution would also have an even numerator.
By graphing x^(x) you can get a better idea of what's going on. The crosses the RHS according to Desmos at around 0.3261
https://www.desmos.com/calculator/prvzcru2ut
A pretty good approximation of this decimal is 14/43 which you can see is pretty close to a solution.
If the actual solution is irrational, we can get arbitrarily close to it in this manner. I doubt there is an exact rational form as for the positive soltuons.

Posted by Jer
on 20231015 12:24:30 