In x + In(x-1)=1
x>0
x-1>0 x>1
\(ln[x(x-1)]=1\\ ln[x^2-x]=1\\ e^{ln[x^2-x]}=e^1\\ x^2-x=e\\ x^2-x-e=0\\ x=\frac{1\pm\sqrt{1+4e}}{2}\\ \text{but x>1 so}\\ x=\frac{1+\sqrt{1+4e}}{2}\\\)
check
LHS
=ln ((1+sqrt(1+4e))/2) + ln(((1+sqrt(1+4e))/2)-1)
= 0.9999999999999998
=RHS
