TITLE: Is $\ln (x/y) = \ln |x| - \ln |y|$?
QUESTION [6 upvotes]: If I am to find the $y$ of the following equation
$$\ln(x/y) = x + y,$$
you would first try to split the left-hand side by doing $\ln x - \ln y$.
My question is, do you need to put the absolute function over both $x$ and $y$?
I think although you know that $x/y > 0$, you don't know if both are bigger or smaller than zero. In my opinion I don't think it's mathematically correct since both $x$ and $y$ could be negative. So I'm wondering if it's more correct to write like this instead:
$$\ln |x| - \ln |y| = x + y.$$
Could anyone let me know if this is indeed the correct way to split the log? Thanks!

REPLY [6 votes]: The expression $\ln(x/y)$ only makes sense when $x/y>0$. In this specific case we have, $x>0$ and $y>0$, or, $x<0$ and $y<0$. When ($x>0$ and $y>0$) or ($x<0$ and $y<0$), we can write,
$$\frac xy=\frac{|x|}{|y|}$$
and then,
$$\ln (x/y)=\ln |x|-\ln |y|.$$