Luck and logic to me, is one of those series that knows what it wants to be, in that it is not supposed to be all that groundbreaking in any sense. Its a nice small show that is heavily cliched, with entertaining action and extremely easy to follow plot/story...
There's not much to go on about here. Good guys defend the city from bad guys that come through these gates from hell or something like that, sprinkle all of this with personal dramatic character plots (each character gets an episode) and top it off with the biggest bad they have to fight at the end. Like i said, easy to grasp.
The art in general is standard fair, though there is quite a heavy use of 3D for the action scenes of which some might find jarring, it is kind of strange seeing the characters so still in 2D and then very kind rubbery ragdoll movement with 3D.
There are some 2D scenes where you can see that the animators screwed up, like one scene at the end of episode 3 i think, it looked like the game was having server issues, but then i realized that this wasn't a game, it was an anime.
The character designs were nice, all of the transformations were pretty neat which nicely fitted each of there personalities.
Some of the action scenes can seem rather lazy though, with the typical "character performs attack shot" and then switches to "character getting hit shot".
Personally, I quite like the BGM, I might even hunt it down somewhere soon. Its mostly the action scene BGM I like with its sci-fi orchestralish sound. I really like the OP too, it kept me from skipping it every episode, The ED however is not to my taste.
The characters are not very interesting, they are very cliched and i will list the main team of them here.
Yoshichika tsurugi - The main character of the series and... well... thats it... he IS a main character. uses a shield as his weapon (tank).
Tamaki Yurine - The intelligent leader type girl, uses a controlled ribbon thing and a wand as a weapon and also heals (healer)
Chloe Maxwell - The energetic arrogant girl, uses a flaming broadsword and often charges into battle without second thought (DPS)
Mana Asuha - The quiet mysterious girl with a troubled past, uses guns, mainly a sniper, and also sneaks around by turning invisible n stealthy n stuff. (support/ranger)
The characters are all teamed up with goddesses that combine with them to transform, but these goddesses are the most disappointing characters in this series since there nearly completely shoved aside and we learn next to nothing about them. I'd say i wouldn't even notice if they were removed from the series altogether.
For what it is, I really enjoyed this series, it was something i could sit down and watch without getting so invested that it would consume my life. Its a cliched "save the world show" and it knows well. As a finished series, i would recommend this to anyone who's got nothing else to do and just wants a quick anime fix rather than re-watching a presumably better series.