A friend asked me if i could sove Ln(x)=-x in exact form.
The solution i gave was x=e^(-e)^(-e)^(-e)^...
Whilst exploring this answer i have had difficulty believing this to be well defined or even correct.
A similar problem is x^x^x^x....=2 (or 4)
Which can be solved as x^2=2 x=root 2
or x^4=4 x=root 2 also
I would apreciate any thoughts on this and infinite exponention. All proofs are welcome.