It's not his fault if he has to go! If you don't take him out then you'll have a mess to clean up in the morning. He will eventually sleep through the night, but if he's not ready yet then there's nothing you can do - except let him out.
It sounds like there is something more going on here than lack of sleep. Of course I get annoyed when Loki needs to go out at night, but it's not his fault. He has a medical condition so he has to go a LOT so I am very proud of him and the fact that he is house trained - but it took A YEAR of constant training to get there. Sometimes he needs to go out every 15 minutes, even during my favorite TV show... thank goodness for the DVR
We don't even have kids but the dogs wake us up constantly at night. They get up to bark at the neighbor's loud truck. They kick me when they are running in their dreams. Sometimes they get sick at night. Sometimes Sammy walks across me to find her perfect spot ON my pillow. No matter what they get up at 6:30 for breakfast even on Saturday. We haven't slept through the night in 2 weeks after Sammy's knee surgery. When they wake us up we get up, figure out what they need cuddle them and go back to sleep. Their sighs and kisses are enough thanks when they settle back in. I really hope that you get to experience that, rather than just frustration. Some frustration is normal but it should come and go, in between all the awesome stuff. It's all worth it as they lay here sleeping on my feet...
Have you tried signing up for classes??? That's how I learned how to communicate with Loki and he has a HUGE vocabulary now and knows so many commands. I have to spell things as I can't even say "I'm going to walk out to the mailbox" to my husband without Loki jumping out of a deep sleep and waiting for me by the door to go for his "walk" You and Logan might appreciate each other a little more if you went to a few puppy classes!
Oh - another thought. Do you know WHY he is waking up? I find that once mine are up we pretty much have to take them out or they will beg at the door to go out and won't come back to bed, even if WE wake them up. Like if I get up to go to the bathroom at night and disturb them by turning the light on or whatever, they will get up and want to go out. Maybe his bladder isn't waking him up. Maybe it's a neighbor or something else. Try covering his crate or experiment with different beds (he doesn't sleep with you, right?)