complex analysis - Why do we say the principal branch of the logarithm has the negative axis removed?

My question is really simple. I didn't understand why do we say the principal branch of the logarithm has the negative axis removed (the branch cut). Usually, the argument of the principal branch of the logarithm without the branch cut is defined on $\{z:-\pi\lt \arg (z)\le\pi\}$. So the negative axis is still there since we are allowed to use $\pi$ as argument.

Could someone clarifies me this?