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 $\log_{-2} x$  ?
 A: First off, there is an error in the textbook, as you can have the base of the logarithm less than 1 and greater than 0. The based of the real logarithm is generally $b\in(0,\infty),b\neq1$. As for negative bases, we experience some weird problems and inconsistencies. For example, it would be logical that
$$(-2)^2=4\implies\log_{-2}(4)=2$$
However, using the change of base identity,
$$2=\log_{-2}(4)=\frac{\log_2(4)}{\log_{2}(-2)}\implies\log_{2}(-2)=1\implies2^1=-2$$
which is obviously a contradiction. 
Furthermore, we lose some pleasant properties such as continuity in the real plane. For example, it might be logical to say that
$$(-27)^{\frac13}=-3\implies\log_{-27}(-3)=\frac13$$
but an approximation to $\frac13$ to $n$ decimal places is $\frac{10^n-1}{3\cdot 10^n}$ (you should be able to see why), and
$$(-27)^{\frac{10^n-1}{3\cdot 10^n}}=\left((-27)^{10^n-1}\right)^{\frac1{3\cdot10^n-1}}\notin\mathbb{R}$$
since $10^n-1$ is odd, $3\cdot10^n$ is even, an odd power of a negative number is negative, and an even root of a negative number is not real. So while an exponent of $\frac13$ has a well-defined real root, the area close to it does not. It is similar with other powers and negative bases, and you end with a disjointed set of points on the real axis. Since the logarithm is the inverse of multiplication, we similarly get a mess of single points where it is defined.
So to preserve our log laws and maintain a smooth graph, it is ideal to restrict the base of the logarithm to avoid negative values.
A: 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.
A: In principle there is nothing wrong with this, it is an analytical function on the complex plane. However for real numbers you will run into difficulties, for example what is $\log_{-2}(2)$? Certainly not real. 
A more interesting question may be why cant the base be in the interval $(0,1)$? You should find that if we extend our definition to include this interval then $\log_{1/b} = - \log_b$, so often we would just use the basis that is in the interval $(1,\infty)$. However, as  has been pointed out in the comments, this is no reason to remove $(0,1)$ from the definition and so it is ok to use if you genuinely feel it is best. 
