So when you integrate to get a log you need to have absolute value signs.
Why is that?
Obviously you can't have a negative log, but why is this correct as opposed to just defining that x>0 in log(x)?
Well, for one thing, you can't always require that x be postive!
For example, find the area bounded by the graphs of y= 1/x, y= 0, x= -2, and x= -1.
That is [tex]\int_{-2}^{-1} (0-\frac{1}{x})dx= -\int_{-2}^{-1}\frac{dx}{x}[/itex].
Now, we have $\displaystyle -ln(|x|)|_{-2}^{-1}= -(ln(|-1|)- ln(|2|)= ln(2)$
That is the same as we would get for the area bounded by y= 1/x, y= 0, x=1, and x= 2, which, because of the symmetry is what we should get.