In this simple integral why is the constant natural logged? I'm reading through some lecture notes and see this in the context of solving ODEs:
$$\int\frac{dy}{y}=\int\frac{dx}{x} \rightarrow \ln{|y|}=\ln{|x|}+\ln{|C|}$$ why is the constant of integration natural logged here?
 A: In this form, it is evident that you can rewrite the result as $\ln \left| C x \right|$.  Perhaps this is less evident from the form $\ln |x| + C$.
A: Normally, you need the boundary values to solve the ODE. Assume $y(x_0)=y_0$, then the solution is,
$$\ln |y| - \ln|y_0| = \ln |x| - \ln|x_0| $$
Thus, $\ln|C|$ is necessary and is to be determined via,
$$\ln |C| = \ln |y_0| - \ln|x_0|=\ln|\frac{y_0}{x_0}|$$
A: No real reason, from this simple equation, that I can see. It could be any $C$. Perhaps the author intended to take the exponential of both sides in the following step and remind you that in this case the constant term must be non-negative.
A: See this:
$\int\frac{dy}{y}=\int\frac{dx}{x} \Rightarrow \int\frac{dy}{y}=\int\frac{dx}{x} + c$
$\ln |y| = \ln |x| + c \Rightarrow  \ln |y/x| = c \Rightarrow  y/x = \pm e^c \Rightarrow   y= \pm e^c x \Rightarrow y = C x $ [Where C is non zero constant.]
$\Rightarrow \ln |y| = \ln |x| + \ln |C|$
Instead of doing this much: 
One prefers to write in short as: $ \ln |y| = \ln |x| + \ln |C|$
