At first glance, the intuitive answer is simply \(\displaystyle \ln (x-1) - \ln (x+1) = \ln\frac{x-1}{x+1}\), however computer algebra systems give me a slightly different answer: \(\displaystyle \ln (1 - x) - \ln (x+1)\).

Obviously, the first term was negated. I assume this has something to do with the fact that logarithms only have real outputs with non-negative inputs.

Could someone please explain this?