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?

In the special case when the given number and base both are real negative, log is well defined in terms of complex numbers and there are no restrictions.
$$ \log _{-a}(-N)=\frac{\log (-N)}{\log (-a)}= \frac{\log (N) + i \pi}{\log (a)+ i \pi} $$
$$=\frac{\pi-i\log (N)}{\pi-i\log (a)}$$
which is a complex number. The new unlabeled base can be again any real number b.
EDIT1:
If new base $b$ is positive then there is no problem but if negative we again have
$$ b= - |b|,\quad \log _{\;b} N =\frac {\log N }{\log b}=\frac {\log N }{i \pi+ \log |b|} $$
and the nestings / branch cuts of log can  propogate continuously depending on the sign of newly chosen/introduced bases.
