The purpose of this paper is to prove the basic law of information and its application in logic, mathematics and translation by expounding the ecological characteristics of information and its scientific research. The first is to explore the essence of information from the diversity, and then to explore the way of information science research from multiple perspectives. Finally, through the social-ecological characteristics of the phenomenon information, at the same time, through the logical-mathematical characteristics of essential information, Ultimately, reveal the basic law of information. The result is that the basic law of information and its application is proven in logic, mathematics and translation. The significance of the phenomenon is that the characteristics of the phenomenon information are highlighted in a variety of forms. The existence of the essential information is known in the order of its order and position in the generalized text gene and its combination form system. Since the machine translation is essentially the translation memory in the vocabulary , Phrases, sentences, paragraphs and chapters are more satisfied with the bilingual pair of levels, therefore, they are in logic, mathematics and language in three aspects to follow the basic principles, namely: this article reveals the three basic laws of information.