For context, I am looking back at an old question regarding a curious family of recursively defined logarithm towers. What slightly unsettles me is the "discrete jump" between each curve as $k$ increases. I want to be able to patch it together by continuity in some sense.
The natural logarithm is well known mathematical function which for example is the functional inverse to the exponential function.
But how can we find or calculate a fractional logarithm function? $$\log^{1/k}(x) \text{ s.t. } (\log^{1/k} \underset{k\text{ times}}{\underbrace{\circ\cdots \circ}} \log^{1/k})(x) = \log(x)$$
We also want it to be "maximally smooth" in some sense.
Here is one approach without smoothing so far, but which seems to work well enough on small intervals.
We consider $x\to \log(1+x)=f(x) : x \in [0,1]$
Represent function decomposition as a matrix $M_f$ somehow.
Make guess of $M^{1/k}$, we make guess $M^{1/k} \approx M_{\frac{1}k f + \frac{k-1}k x}$ in other words a linear weighting between $f(x) = x$ and $f(x) = \log(x+1)$
Calculate error $e = |(M^{1/k})^k-M|$
Update guess based on $e$. Iterate from 3 until error "small enough".
Here is an approximation for $\log^{1/8}(1+x)$ :
We can see the uncertainty spreading with the matrix power. But the mean value hits real close to true. Max pointwise error circa $10^{-6}$. After some tuning and larger matrices we can expand to a larger domain:
Error residual : relative error below 1 per million.