Doesn't the constant matter in $\int\frac{1}{x}dx=\ln(kx)+C$ instead of writing $\ln(x)+C$?

Question

If I know that $\frac{d}{dx}\ln(5x)=\frac{1}{x}$ then shouldn't I be taught that $\int\frac{1}{x}dx=\ln(kx)+C$ instead of $\ln(x)+C$? Is there a reason why we don't care about the constant inside the $\ln$ function?

Answer 1

$\ln kx=\ln k+\ln x$ and $\ln k$ is a constant so it doesn't matter.

Answer 2

The constant matter in $\int 1/x dx=\ln\ (kx)+C$ instead of writing $\ln\ (x)+C$ and this be wrong, because there we use two constants, namely $k$ and $C$. This integral formula generally a 1st order differential eqn, since $d/dx(\ln\ x)=1/x$, i.e, $dy/dx=1/x$ So, $\int 1/x dx=\ln\ k+\ln\ x=\ln\ kx$ and here $\ln\ k$ is a constant. It is the correct rule.