I have seen two justifications for the absolute value sign in ln|x| as the antiderivative of 1/xm, but neither one seems sufficient. The first one is quite lame, that ln can only deal with a non-zero positive domain (as long as we are sticking to the real numbers). But this would not rule out a definition such as (as example only)
ln(x) if x is positive
-ln(|x|) if x is negative.
Or something like this. I am not proposing this as a definition; only showing how the justification above is insufficient.
The next justification I have seen is that the area under the curve 1/x over an interval (a,b), with a<b<0, will be the same as the area under the curve 1/x over the interval (|b|,|a|), so we take the absolute value. However, the integral is not the same thing as the area; if we are looking at the negative side of the x-axis, we get negative "signed areas" from the integral, not the areas.
So, why the absolute value? Thanks.