Originally posted by ChrisM Could it simply be a dirty electronic contact? How hard is it to remove the top?
Most likely the meter switch that is easy to access by removing the bottom plate:
Does the Pentax MX have mirror lock-up? and repair tips
Scroll down and look at the second picture, where red arrow points. That is the meter switch and cleaning it should help. With removed bottom plate try operating shutter release button and see how it works, it is simple and easy to understand. To clean, cut a strip of paper, put it between contacts, press shutter release to sandwich it between contacts and gently slide paper out. Repeat several times if necessary. A drop of alcohol before cleaning helps.
Once cleaned and adjusted (I had to adjust mine) do the following tests: (1) with shutter release button free the LEDs should NOT operate; (2) with rewind lever in extended position and shutter release locked in "metering" position the LEDs should operate; (3) with shutter release in locked position and while applying pressure on it LEDs should NOT operate -- if they do contacts are too close and you'll have to gently flex outer contact plate to allow for a bit more clearance. (Lack of clearance in "L" position is the main cause of battery leakage, as paragraph just before the picture describes.)
If this does not help then shutter speed selector or aperture coupler contacts may need cleaning. However, if your problem is consistent at all shutter speeds and all apertures then it is most likely the meter switch that is easy to fix as described.