logic can't help us

I'm an atheist by logic.

It may surprise you to hear that logic is not a religious system. Logic has nothing to say about religion, one way or another. Logic is a system for evaluating the truth of a statement.
"God exists" is a logical statement, but logic does not tell you whether the statement is true or not.

If the sky is blue, then the sun is out.

This is a logical statement, because it has an antecedent (condition) and a consequence. It is also a true statement.

If the sky is blue, then we will cry.

This is another logical statement, but we do not know if it is a true statement.

If the sky is filled with diamonds, then cows are green.

This statement is logical, but we do not know if it is true unless we know more about the sky, diamonds, and cows. On this planet, and in English, the statement is most likely false.

If green, then run.

Finally we have an illogical statement. The antecedent is not clear. If what is green? Neither is the consequence. Who should run? This statement cannot be evaluated even though we know what green is and what running is.
The statement "Water exists" is logically true because we know what water is. The complete statement would be

If water is a clear, liquid substance that exists in three states and is necessary for life, then water exists.

Science is what helps us to logically deduce that this statement is true. Science cannot help us with the statement that God exists, however.
Science tells us about nature, and God does not exist in nature. To put it another way, the existence of God does not logically follow from what science tells us about nature. If we define nature, or the natural world, as that which can be observed by the human senses (sight, taste, touch, smell, hearing, assisted or not), then science will never tell us about God. Note that science also does not tell us about what isn't in nature. Science cannot say for sure that unicorns or fairies do not exist. All a scientist can say is "I have never observed a unicorn or a fairy in nature". Also, the most a scientist can truthfully say against the existence of God is, "I have never observed God in nature". Logically, this is not equivalent to "God does not exist".
Science can say, "This is a natural process" or "This entity is found in nature" but in order for science to say "God does not exist", it has to change what it examines. Science does not observe, examine, or investigate entities outside of nature, so it can have make no definitive conclusion about God.