Why base of a logrithim function must be greater than one? I'm in domain section of my textbook. It says that for logarithm functions, the base must be greater than one.
I can understand why base shouldn't be one but what is problem with negative numbers?
for example, What's wrong with $\log_{-2} x$  ?

Because when working only with real numbers, we do not know how to make sense of the expression $(-2)^\sqrt{2}$, for example.
In general the expression $a^x$, with $x \in \mathbb R$, makes sense only if $a > 0$.
Since $a$ is the basis of the logarithm, you can see why we limit ourselves with positive base.
With complex number the situation is different; we can always define $a^x = e^{x \ln^\mathbb C a}$, where $\ln^\mathbb C$ indicates the complex logarithm, which is a multi-values function, formally given by 
$$\ln^\mathbb C z = \ln^\mathbb R |z| + (2k\pi + \arg z)i$$
(note though that this way $a^x$ is not a well-defined complex number, it is an infinite collection of complex numbers )
In principle then one could define the complex logarithm for bases different than $e$ (also negative ones!), but it is not useful to do so.
