# Thread: differentiating absolute value graphs

1. ## differentiating absolute value graphs

hi, im a bit confused on the graph of y=abs(lnx)

when i graph it, my answer only appears in the first quadrant since x>0, and where the negative starts reflecting on the x axis is the c.p so i cant differentiate that spot. now when i check my graph on wolfram alpha, it says that the graph is in both the 2nd and 1st quadrant - Wolfram|Alpha

You wrote the function $\displaystyle y=\left|\ln x\right|$ , and this function's defined only for $\displaystyle x>0$, so I don't know who or what is telling you that that its graph is in both the first and second quadrants: it/he/she is wrong.
Since by definition $\displaystyle |x|:=\left\{\begin{matrix}x&\mbox{ , if }x\ge 0\\-x&\mbox{ , if }x<0\end{matrix}\right.$ , we get $\displaystyle y=|\ln x|=\left\{\begin{matrix}\ln x&\mbox{ , if }\ln x\ge 0\\-\ln x&\mbox{ , if }\ln x<0\end{matrix}\right.$ $\displaystyle =\left\{\begin{matrix}\ln x&\mbox{ , if }x\ge 1\\-\ln x&\mbox{ , if }0\le x<1\end{matrix}\right.$
Thus, the graph of this function is a mirror-reflexion thru the x-axis of the graph of $\displaystyle \ln x$, for $\displaystyle 0\le x<1$, and it is exactly the same as the graph of $\displaystyle \ln x$ for $\displaystyle x> 1$ .