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 $\log_{-2} x$  ?
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.
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.