In differential equations, why is ln|x|+c = ln(cx)

Question

So, let's say I get the expression $ln|x|+c, c \in R$. My reasoning is that I can do the following:

let $c_1 = e^c$, then $c_1 > 0$

$ln|x|+c, c \in R$ becomes:

$ln(c_1|x|), c_1 >0$

How I can justify removing the absolute value?

Answer 1

$\ln(|x|)+c$ usually arises when integrating $1/x$ in calculus. $\ln(cx)$ would be just as good as an antiderivative (where you want to let $c$ have the same sign as $x$). It has the advantage of making it not appear like there has to be a connection between the values for $x > 0$ and those for $x < 0$. Also, when you come to deal with complex variables, $\ln (|z|)$ is not analytic while $\ln(cz)$ is.