An approximation of logarithm log(1 + x) with connections to combinatorics or probability? [closed]

I am looking for an approximation of $$\ln(1 + x)$$ or $$\ln(x)$$.

We know $$\ln(1 + x)$$ has a Taylor expansion:

$${\displaystyle \ln(1+x)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}x^{k}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots ,}$$

But for the problem I am working on, (a problem in random walk), I would like to get an alternative approximation of $$\ln(1 + x)$$, and hope it has some natural connections to combinatorics or probability or random walk.

Is there an approximation of $$\ln(1 + x)$$ or $$\ln(x)$$ which has some connections to combinatorics or probability or random walk ?

At it's simplest, $$C(x)=-\log(1-x)= \sum_{n > 0} (n-1)! x^n/n!$$ is the exponential generating function (e.g.f.) for the Joyal species of cyclic permutations in the theory of combinatorial species. The number of ways of uniquely ordering the natural numbers up to $$n$$ on an oriented circle is $$(n-1)!$$.
• It's a generalization, not approx, that pops up in advanced combinatorics, algebra, and topology. Simply replace each term of the Taylor series $x^k/k$ by $(a.x)^k/k = a_k x^k/k$ where $a_k$ is an arbitrary indeterminate. Sep 5, 2019 at 1:02