I asked WolframAlpha to solve a certain differential equation and it gave it in this form: $f(x)=(x-1)(\ln(x-1)-i\pi-1)$. Now I am only interested in this function when $x$ is in the interval $(0,1)$. In this case, I think $f(x)$ will always be a real number.
My question is, is there a way to simplify this expression in the interval $(0,1)$, so that it makes no reference to complex logarithms or imaginary numbers? I just want to express it in terms of real functions.
We assume principal values for the multivalued functions we invoke here. Since $x\in(0,1)$, $x-1$ is negative and $$\ln(x-1)=\ln(1-x)+i\pi$$ Thus $$(x-1)(\ln(x-1)-i\pi-1)=(x-1)(\ln(1-x)+i\pi-i\pi-1)=(x-1)(\ln(1-x)-1)$$