What you said was "Why does lnX!=XlnX-X?". Now you tell us not only did you NOT mean "ln X!= X ln X- X, but you also tell us that you already KNOW the answer to your (unstated) question.
Limit of x - lnx as x approaches infinity: Solving Indeterminate ...
The discussion revolves around the calculation of the integral of the function e^x (lnx) and the anti-derivative of x^x. Participants explore various methods and approaches to tackle these integrals, including the use of series and numerical integration techniques.
How Do I Calculate the Integral of e^x (lnx) and Anti-Derivative of x^x?
Your first approach was correct, the second one, as already pointed, instead of ln (lnx) you took lnx.