Slog(Exponential Factorial(x))

[tommy1729]

EF(x) = exponential factorial(x) = x^(x-1)^(x-2)^...^3^2^1.
What happens if you do the tetration logarithm of the exponential factorial function. (I am thinking tetration logarithm base the Tetra-Euler Number.) How can the tetration logarithm of the exponential factorial function be approximated?
slog(e4,EF(1)) = 0.
slog(e4,EF(2)) ~ .636.
slog(e4,EF(3)) ~ 1.612.
slog(e4,EF(4)) ~ 2.693.
slog(e4,EF(5)) ~ 3.703.
Numbers worked out with the Kneser method.
This is my 16th thread! 
I conjecture that slog(k,EF(x))-x approaches a number, as x grows larger and larger. For any real k greater than eta.

EF(x) = exponential factorial(x) = x^(x-1)^(x-2)^...^3^2^1.
Let slogef(x) be its inverse.

Let HF(x) = hyperfactorial(x) = 2^3^..^x.

The argument I posted earlier is pretty close to a proof that slog(HF(x)) - x does not converge to a constant as x grows.

I suspect (conjecture) that slogef(HF(x)) -x also does not converge.
need to think more about it.

another related conjecture is 

slog(EF( x - ln(x) )) - x converges as x grows ?

Im getting ideas ...


regards

tommy1729