did i make a mistake

crown casino melbourne. Blinds 2/3 cash games. i have over 500 while villian has only 110. game is very loose passive.

preflop im on button with A3. 3 players limit 3. i raise to 15. blinds fold but other players call. flop is 2K5. villian bets out 25. co calls and i call. turn comes a 8 and board is rainbow. villian bets out 25 co fold and i call.

it was pretty clear that he had a K with some decent kicker. i figured that i was getting 5 to 1 to the flop on roughly 7 outs (three aces and four 3s) so roughly 3 to 1 chance of me hitting to the river. The turn he was giving me 6 to 1 odds (with roughly 8 to 1 implied odds seeing he was pretty much pot commited on the river) and i figured i still had a 6.5 to 1 to catch on the river.

Was my thinking process reasonable or did i make a huge mistake?

I wouldnt agree with everything here. You stated that the game was loose passive therefore I do not believe that your Preflop Raise with an Ace and awful kicker is going to win the pot at this point.
Flop comes and now youre already behind hoping to catch a gutshot. Youre not even sure if the Ace hits you would be ahead. Someone could have A better kicker or A2, A4.
Im not even going into your math. It's just a bad idea to Raise preflop with that kicker and then have no other plan but to chase it all the way to the river.

Personally, I wouldn't have even played ace rag, not to mention raising pre...wait for a better hand.