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 $\log_{-2} x$  ?

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. 
