Argument from logic
1. Minds exist which have and use objective logic.
2. Objective logic cannot be based on our subjective minds, a non-static universe or immaterial abstractions outside of a mind.
3. Objective logic depends and can only derive from a pre-existing necessary first mind with objective logic.
Without laws of logic, we could not make an argument.
We can make an argument.
Therefore, there must be laws of logic.
While this argument is circular, it is a non-fallacious use of circular reasoning. Since we couldn’t prove anything apart from the laws of logic, we must presuppose the laws of logic even to prove they exist. In fact, if someone were trying to disprove that laws of logic exist, he’d have to use the laws of logic in his attempt, thereby refuting himself. Your non-Christian friend must agree there are certain standards that can be proven with circular reasoning.
The basic presupposition—God exists and has revealed Himself in His inerrant, authoritative Word—is the ultimate standard. Presupposing God exists to argue that God exists is a reasonable circular argument because without the God of the Bible, we have no basis for assuming the laws of logic and their properties, let alone absolute morality or the uniformity of nature.
Self evident knowledge exists. For example, we intuitively know about truth, and principles of logic. This describes what we know, innately. The question is, how do we know it? Where does this self evident knowledge come from? Obviously, knowledge doesn't arbitrarily pop into existence (and if it did, it would be irrational by definition). Knowledge is a product of "mind," and therefore must originate or come about via a rational mind. The Christian worldview offers a simple, logical explanation for how we know some things to be self evidently true: An omniscient, logical mind (God) imparted certain foundational knowledge to mankind made in His image.
But how does the atheist account for self evident knowledge? Quite simply, they can't. In a materialistic or naturalistic worldview, all knowledge is inductive, experiential, and must be gained via a process of thinking. If knowledge could arbitrarily pop into human consciousness (apart from impartation by God), this would mean that knowledge can be gained without a thinking mind. But again, knowledge is conceptual, meaning that it's a product of thinking minds, and therefore cannot come about without thinking minds.
Furthermore, even if knowledge could arbitrarily come about apart from minds that are (already) aware of truth and logic, how would we know that such "knowledge" is indeed true and logical? For example, imagine that the law of non contradiction, which states that a proposition cannot be both true and false at the same time and in the same way, arbitrarily popped into human consciousness apart from any process of reasoning. How would we know that the law of non contradiction is indeed true and logical without logically concluding it via a process of reasoned thought? We simply wouldn't. In this scenario, the law of non contradiction would necessarily be irrational because it originated without reason or rational thought. The law of non contradiction cannot "confirm its own truth value." The truth of a proposition is confirmed by mind, which requires a priori awareness of truth and logic whereby one can engage in reasoning about the proposition. Therefore, the materialistic is stuck with a classic catch 22. To validate the truth of propositions about truth and logic would require a priori awareness of truth and logic - whereby the truth value of such propositions could be concluded via logical thought processes. In other words, to conclude the truth of the laws of logic would require knowledge of true laws of logic to begin with! One would already have to know what they're "attempting to know."
Finally, it is viciously circular and therefore logically impossible to determine the validity of one's own logic and reasoning via one's own logic and reasoning (like trying to check the accuracy of a homemade ruler by using the same ruler).
If the atheist indicates that self evident truths do indeed exist, they've unwittingly presented a clear defeater for their own worldview. The "how" of self evident truths can only be rationally explained via a transcendent, omniscient mind - the mind of God.