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

[marraco]

I think \( a_1=ln(a) a \) makes sense.

Regards

Tommy1729

For bases b > 2 i think the following makes sense :

For x > 0 :
F " (x) > F ' (x)/2
F ' (x) > 0
And
F ' (0) = b/2

What do you think ?

Seems arbitrary to me. Do you care to expand?

There is probably some value for a₁ that provides some benefit. For example, some values makes \( \\[15pt]

{^xa\approx ln_{N_a}(a^{a^{x-1}})} \) (which is right bracket tetration). Other values make \( \\[15pt]

{^xa\approx ln_{N_a}(^xsroot(a))} \).

I don't like a₁=0, because it makes most bases look weird.
For example, this is tetration base \( \\[15pt]

{e^{\frac{1}{e}}} \), when a₁=0 (up), and when there is no restriction in a₁ (down; It gives a₁=0,6118~~~~~)



I find the smoothed curve more appealing, because I suspect that flow in pipes is governed by tetration.

This is a semi empiric "Moody diagram" of flow in pipes. To me, it resembles tetration. When flow turns fully turbulent, it converges to a constant value.