Question: "How should a Christian view logic?"
Answer: Logic is the science of deriving truth through the analysis of facts either directly (deductively) or indirectly (inductively). Logic takes given presuppositions, analyzes relationships, compares them with other known factors, and arrives at a conclusion that identifies a previously unknown fact. Logic is math with ideas instead of numbers. It is a way of identifying the relationships between ideas.
Logic appears to be one of the natural laws God put into place at the creation of the universe. Then, God created mankind with a mind and the ability to reason. Being a creation of God, logic is a good thing which, when used properly, can point us toward God. Unfortunately, it is easy to use logic incorrectly.
The science of logic deals with the relational formulae of ideas. Like numbers in math, ideas can be plugged into formulae that show their relationships with other ideas. It is beneficial to understand the basics of these formulae. Modern arguments are often saturated with emotion, which can stymie conversation and preclude a useful resolution. Passion can impede the path to truth. Often, truth is hidden by what is known as fallacy—argumentation based on false logic and erroneous reasoning. Fallacy is a bullying tactic, and it doesn't lend itself to profitable discussion.
Logic in a practical sense includes both the formulae and the facts. The formulae provide the relationships, but there must be basic ideas available for the formulae to analyze. Although relativism chips away at even the most basic assumptions, most people still rely on empirical evidence—data they accumulate through their senses. Most people are confident making statements such as "I exist" and "the table exists." Logic takes such data and derives further truth. "Anything that has a beginning must have been made by something else" is a logically deduced statement. Further analysis leads to more complex truths, such as “God exists.”
Unfortunately, many debaters inadvertently fall into fallacy because they do not start at the beginning. That is, they allow a pre-conceived, unproved notion to stand in for a fact. Evolutionists start with naturalistic evolution as the basis for their arguments because they do not accept the possibility of miracles. Many religions reject that Jesus is the God-man because they start with Gnosticism (the physical is evil; the spiritual is good). Secularists who insist that religion is an instinctive response to the fear of death start with the assumption that God does not exist.
The truth is, most people are not going to be significantly influenced by logic to believe something contrary to their convictions. Usually, sentiment trumps logic. And, although neither Jesus nor the apostles were strangers to logic, it was not their primary tool. When Peter says to be "ready to make a defense to everyone who asks you to give an account for the hope that is in you" (1 Peter 3:15), he didn't mean to start with the ontological argument for the existence of God. He meant to be ready with the story of our own relationship with God and the hope that has come from it. Someone who bases his beliefs on emotion will not be able to track a logical conversation. Logic in the hands of a trained apologist is a powerful tool. But equally convincing is the "empirical evidence" of the Christian life. We are “the light of the world” (Matthew 5:14); the darkness may not like the light, but it cannot deny its existence. “In your teaching show integrity, seriousness and soundness of speech that cannot be condemned, so that those who oppose you may be ashamed because they have nothing bad to say about us” (Titus 2:7-8).