Jim Gannon handed Gianluca Havern his County debut as the Hatters boss shuffled his pack ahead of next week's play-off semi-final.
He also included three more youngsters, Amari Morgan-Smith, James Vincent and goalkeeper Ben Imeson on the bench.
The first chance of the game fell to the feet of Glenn Poole who launched a blistering strike from the edge of the County area that had beaten Logan but bounced to safety off the foot of the post.
The new-look County side were making some noise at the other end and the hungry Adam Proudlock fed a telling ball into the area that Hamer did well to gather.
David Poole was obstructed on the edge of the area on the 13th minute, Michael Rose took the resulting free-kick but unlike him, it went well-wide.
It was shaping up to be one of those games that typify an end of season affair, with no pressure on either side.
A fresh Leon McSweeney went close on the 20 minute mark, but Hamer cleared the ball with ease. Hamer missed the resulting corner, but unfortunately for County there was no one to provide a decisive touch. McSweeney won another corner moments later, but again County were wasteful.
At the other end Alan Connell tried to do a 'Sodje' when he hid behind Logan, prior to him taking a free-kick. Logan had clearly learnt from earlier mistakes and noted the blonde striker lingering behind him, prior to taking the goal-kick.
Leon McSweeney went close again with a bursting run into the area, but Hamer made the vital block at the near post as the Irish player lost ground.
Michael Rose had another free-kick from the edge of the area as the game started to head towards half-time.
Ryan Lowe and David Poole created another good opportunity for The Hatters when Lowe flicked on a ball for Poole who burst into the area, but Poole ran out of space as another chance went begging.
County got a well-deserved goal on the 50th minute when young Gianluca Havern volleyed home from the edge of the area.
James Tunnicliffe got a round of applause when his skillful run forward provided the highlight of what was becoming a rather relaxed end of season affair.
Substitute Amari Morgan-Smith went close to getting a debut goal as The Hatters looked to double their lead, but County had Logan to thank late on when he twice denied Gary Smith before Ricky Shakes missed a glaring chance late on with a poorly-directed header.
County had to thank Logan for preserving their lead when the Irishman made a brilliant double save to deny Gary Smith in stoppage time.
