thermodynamics - Definition of the entropy

In physics, the word entropy has important physical implications as the amount of "disorder" of a system. In mathematics, a more abstract definition is used. The (Shannon) entropy of a variable $X$ is defined as

$$H(X)=-\Sigma_x P(x)log_2 [P(x)]$$

bits, where $P(x)$ is the probability that $X$ is in the state $x$ , and is defined as $0$ if $P=0$.

Question: Can anyone explain to me why "disorderness" of a system defined as this? especially, where has "$log_2$" come from in this formula?