Improving convergence of Andrew's slog

[quickfur]

[...]
In the above formula let \( f(x)=b[4]x \) for some base \( b \), then the equation is \( f(x+1)=b^{f(x)} \).
Now we can perturb \( f \) in the following way let \( \phi(x+1)=\phi(x) \) be a periodic analytic function (for example \( \phi(x)=\sin(2\pi x) \)) and consider
\( g(x)=f(x+\phi(x)) \)
then
\( g(x+1)=f(x+1+\phi(x+1))=f(x+\phi(x)+1)=b^{f(x+\phi(x))}=b^{g(x)} \) is another analytic function satisfying this equation (and every other solution is a perturbation by a periodic function).
If you choose \( \phi(x) \) low in amplitude the function \( g(x) \) even remains strictly monotonic increasing.

Interesting. This argument also works if applied to exponential functions, using the property that \( e^{x+1}=e\cdot e^x \). In other words, this property is not sufficient to uniquely determine a unique (up to scalar multiple of the exponent) exponential function.

Now, the traditional way (or at least, one of the usual ways) of defining the exponential function is by its inverse: \( L(x+y)=L(x)L(y) \). This gives us a logarithm function unique up to change of base. This seems to suggest that perhaps we need to discover a property of the tetralog (tetrational inverse) function that will lead us to a unique (up to change of base) tetration. Unfortunately, the relationship of tetration to lower operations is non-trivial at best, and it's not obvious which property should be used as our basis.