Why $r$ should be $\in\Bbb Q$ why not $r\in\Bbb R$

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 $r\in\Bbb R$

• As long as $x>0$, $r$ can be any real number , not only a rational one. – Peter Jun 29 '18 at 18:01

In the terms you put it, there is really no problem: that identity holds for all $r\in\Bbb R$.
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 $\frac 1x$
• "The only reason I can see to restrict to $\Bbb Q$ is if you haven't defined $x^r$ for irrational $r$." Or you have defined it for irrational $r$, but proving the property involves quite a bit more machinery than it does for rational $r$. – Arthur Jun 29 '18 at 18:45