When I look up for properties of the natural Logarithm I found in particular this property
$$ \ln(x^r)=r \ln(x) $$
with $$x\in \Bbb R^{+*} $$ and$$ r\in\Bbb Q$$
My Question is : Why $r$ should be $ \in\Bbb Q$ why not $r\in\Bbb R$
because i can't figure out any problem with being $ r\in\Bbb R$
On
It should be for all real $r$. The only reason I can see to restrict to $\Bbb Q$ is if you haven't defined $x^r$ for irrational $r$. You can define it for rational $r$ based on the definition for integers and the laws of exponents. Wherever you saw this may not have made the definition for irrational $r$ yet. That usually goes through defining the exponential function or the natural logarithm, with the natural log defined as the integral of $\frac 1x$
In the terms you put it, there is really no problem: that identity holds for all $r\in\Bbb R$.