No, the laws of logic do not contradict miracles. Before we get into why, let's define our terms.
The laws of logic are rational statements that are universally true. The first law of logic is the Law of Identity which says that something is what it is and is not what it is not. The second law of logic is the Law of Non-Contradiction which states that something cannot be itself and not itself at the same time and in the same sense. The third law of logic is the Law of Excluded Middle which says that statements are either true or false. Such laws and other logical laws form the undergirding foundations of rational thought.
A miracle is an extraordinary event--something that appears to defy the normal laws of physics. Miracles, from the Christian perspective, are authored by God. Examples of miracles would be the parting of the Red Sea, turning water into wine, and the physical resurrection of Jesus.
There is no logical reason why an omnipotent and omniscient God could not part the Red Sea, turn water into wine, and raise someone from the dead. In Christian theology God is the Creator of the universe and the master of all of its laws. Undoubtedly, God would have access to certain laws of the universe that we don't know about and could use them in such a way that they would appear to be miraculous. Also, since he transcends space and time and is not restricted to the physical universe, it is logical to assume that he has characteristics within his own nature that we cannot comprehend. For example, we cannot fathom what it means to be in all places at all time. How does God know all things? How can he be outside of time? We can understand the concepts, but we cannot experience their reality.
So, we can safely conclude that within the Christian perspective of God, he possesses attributes that allow him to access laws of the universe that we do not know about and use them, in combination with his divine power in nature, to bring about the miraculous.
There is nothing illogical about this. Therefore, there's nothing illogical about God performing miracles.