Why bases 0<a<1 don't get love on the forum?

[marraco]

Well there is at least one post; http://math.eretrandre.org/tetrationforu...0&pid=6748 tetration base \( \exp(-e)\approx0.0660 \)

That's were excel gets itchy, because it can't take logarithms of negative numbers. The polynomial gets too close to zero.
I get this



for bases between \( \\[15pt]

{0<b<e^{-e}} \) it remains bounded between 0 and 1 (for x>0), and converges to c, were c is the solution of \( \\[15pt]

{a^{a^c}}\,=\,c \), which seems to have 2 roots, c₁ and c₂, with
\( \\[15pt]

{a^{c_1}\,=\,c_2} \)
\( \\[15pt]

{a^{c_2}\,=\,c_1} \)
\( \\[15pt]

{{c_1}^{c_1}\,=\,{c_2^{c_2}\,=\,{a^{c_1.c_2}} \)
\( \\[15pt]

{a\,=\,{c_1}^{\frac{1}{c_2}}\,={c_2}^{\frac{1}{c_1}}} \)

I suspect that this relation is the key to solve tetration equations:

\( \\[15pt]

{a^{c_1}\,=\,a^{a^{c_2}}\,=\,a^{a^{a^{c_1}}}\,=\,... \)

Here is tetration base a=0.01:
c₁ = 0,941488369
c₂ = 0,013092521


The negative axis probably converges to a real function akin to a cosine.

I need something better than excel.