It can be solved via derivatives and intersections of curves. But how would you solve it in a purely algebraic manner using the W Lambert function?
In principle yes it can, the equation can be transformormrd to:
So formaly we have:
Now the problem is that there a multiple (real) branches of the Lambert W in
the region around , and you are going to have to
evaluate two of them to get all the solution to this. My LambertW calculator
However just plot the functions and you will see that there is another
solution near Numerical methods will find both solutions
once you know where they are.