You are definitely honing in on the issue.

I don't know where you are getting the notion that ln(x) should be ln(|x|). The natural log of a negative number is undefined if you are working with reals and it is a complex number if you are working with in that domain

Well, that's why I told that the argument of ln should be greater than zero. I assumed that the TS problem was to be treated within the reals. Didn't even thought about going into complex. So, ln (x), within the reals, or we use ln(|x|) or we have to include the x>0 restriction. That's the idea I have. But But it's gone 2 years since I solved a bunch of problems regarding logarithmic functions!

 

But one of the points I was trying to make is that you can't just go and use ln(|x|). If you are integrating specific functions then you can use symmetry arguments to write the results in terms of ln(|x|), but that in no way defines the value of ln(x) for x<0.

 

Ok, agree with that!