Change of base formula for Tetration

[bo198214]

\( {}^{x + slog_b(a)}{b} = \exp_b^{[x]}(a) \)

Let me make a short unification. The slog is simply a solution to the Abel equation for \( F(x)=b^x \):
\( f(F(x))=f(x)+1 \)
which is merely our initial condition written for \( f=\text{slog}_b \) instead of \( f^{-1}(x)={}^xb \).

It is well-known that this corresponds to a fractional iteration via
\( F^{\circ t}(x)=f^{-1}(f(x)+t) \)
Moreover that any fractional iteration must be of this form.