about power towers and base change

[sheldonison]

....Im fascinated by this f(x).
My x_n is now given by f^[-1]( exp^[n] f(x) ).

SO x_n does grow superexponential !

Thank you Sheldon !

you're welcome, and you are correct.
The approximate value for b^^n = e^^(n+2)
as n to infinity, is b~= 302263.280461758

2 remaining questions pop up immediately in my head :

1) how fast does f(x) grow ? Can we express f(x) in other functions ?
2) is there a limit form for f^[-1](x) ??

This is the equation for f
\( f(x) = \lim_{n\to \infty} \ln^{[n]} x \uparrow\uparrow n \)

For numerical results for \( f^{-1}(x) \), I iterate using Newton's approximations and it works pretty well.

for n=2, you get a sense of how f(x) grows:
\( f_2(x) = \ln^{o2}(x\uparrow \uparrow 2) = \ln(x) + \ln(\ln(x)) \)
for n=3
\( f_3(x) = \ln^{o3}(x\uparrow \uparrow 3) = \ln(\exp(f_2(x))+\ln(\ln(x))) = f_2(x) + \ln(1+\frac{\ln(\ln(x))}{\exp(f_2(x))}) \)
for n=4,
\( f_4(x) = \ln^{o4}(x\uparrow \uparrow 4) = f_3(x) + \ln(1 + \frac{1}{\exp(f_3(x))}\ln(1+\frac{\ln(\ln(x))}{\exp^{o2}(f_3(x))} )) \)
.... for n=5, there are three nested logarithms ....
- Sheldon