Why logarithm of a negative number to a negative base is not defined? I have studied that the numbers of the form N=aˣ can be written as $\log_aN=x \;\forall N>0, a>0, a\not=1.$
Let's take $N=-8$ and $a=-2$ then $-8=(-2)^3$ and $\log_{(-2)}-8 = 3$.
Why there are restrictions on the number and the base even though $\log_{(-2)}-8 = 3$ is defined? Is there something I'm missing as I'm only learning the propaedeutics of mathematics?

If $a>0$, then $a^x$ is defined for every real number $x$ and, if $y>0$, what we denote by $\log_ay$ is the only $x\in\Bbb R$ such that $a^x=y$.
But if $a<0$ then $a^x$ is defined only for some numbers $x$; for instance, $a^{1/2}$ is undefined. That's why we don't talk about $\log_ay$; it's undefined for most numbers $y$.
