Logic consists of words arranged in certain patterns. Words can be arranged so as to do all kinds of things that do not reflect reality
I am the queen of the universe and everything that happens is because of me.
I just arranged some words in a logical pattern. Do those words reflect reality? (Now you get to say "no".)
You know those apologetic pseudo-scientific pseudo-philosophical stories that go around the internet? ("Darkness is the absence of light, bad is the absence of good, and sin is the absence of god..."). Same stuff going on. Words that might sound nice but that don't actually reflect reality, just like fairy tales, myths, jokes, and fictional stories.
Words only state something
; and that something
may or may not be true, valid, real. Saying something does not prove
anything. Any religion--Mormonism, Scientology, Jehovah's Witnesses, Nation of Islam-- can convince some people to believe in it by using logical-sounding language to package human experiences in a particular way. That, again, is not proof of anything.
(Except maybe the gullibility of people.)
You cannot "prove" the existence of a god with logical-sounding words. And you should not have to. You should be able to clearly demonstrate the existence of a god
, not just keep restating that god must exist because of this nice feeling you have, or because of that strange event that happened once, or because of that especially nifty piece of irrefutable logic.
Fact: if there was any sentient being powerful enough to create life or the entire universe, and that being wanted every human to know about it, we all would.
Consider the sun. We all know that there is a sun. We don't need to use ancient stories, logical arguments or apologetics to "prove" that the sun is there. We can see it, feel its rays, photograph it, document and measure its effects on the planet in a million different ways that all give the same results.
Imagine if we had thousands of different competing ideas about the sun, all with equally fervent supporters, articulate leaders, bodies of literature. And none can produce any evidence that what they say is true. You just have to pick a group that is popular in your region, and have faith
that they are right..... 
If there is a powerful, loving god, it should be at least as clearly evident as the sun. There would be no way possible to not
know about it! There should be no opposing arguments, no doubters, no wavering faith, no atheists, just as there is no group of humans who doubt the existence of the sun.
There should be no need to base your belief on ancient manuscripts with pointless passages that contradict each other, and vague prophecies that only make sense in retrospect. There should be no need for complicated apologetic excuses for why god does not seem to exist. There should not be a need for "faith" to cover for the fact that everyday life experiences make it seem like god is not real. Most definitely there should be no need to "look into your heart" to find out if god exists.
Do people need to "understand the right interpretation of the sacred text" or "look into their hearts" or "have faith and maybe you will know the truth when you die" about the sun? Or about anything else commonly accepted as real besides supernatural sh!t?
The sun exists. Gods don't. Case closed.