Approximate ln(1-x) under strict conditions

Question

As title, what's the approximation of $\ln(1-x)$ in terms of only $\ln(x)$, its linear combination (i.e. $a\ln(x)+b$) or $(\ln(x))^2$ ,where $x$ is between 0 and 1?

Answer 1

This is not an analytical answer to your question.

I used a nonlinear Regression with the least squares error measure. For $\ln(1-x)$ approximated by $a\ln(x)+b$ I received $a\approx 0.19838137...$ and $b \approx 0$. But the fit is very poor.

The problem is partially caused as you are trying to approximate a decreasing function by an increasing function. Even a linear approximation would work better.