−ln22−ln22=∓ln22 − x ln 2 2 e − x ln 2 2 = ∓ ln 2 2
Here we can apply a function known as the Lambert W function. If = x e x = a , then =() x = W ( a ) .
−ln22=(∓ln22) − x ln 2 2 = W ( ∓ ln 2 2 )
=−2(∓ln22)ln2 x = − 2 W ( ∓ ln 2 2 ) ln 2
For negative values of x , () W ( x ) has 2 real solutions for −−1<<0 − e − 1 < x < 0 .
−ln22 − ln 2 2 satisfies that condition, so we have 3 real solutions overall. One real solution for the positive input, and 2 real solutions for the negative input.
I used python to calculate the values. The dps property is the level of decimal precision, because the mpmath library does arbitrary precision math. For the 3rd output line, the -1 parameter gives us the second real solution for small negative inputs. If you are interested in complex solutions, you can change that second parameter to other integer values. 0 is the default number for that parameter.