Is it my battery, voltage regulator or ???

[hgs_notes]

Sunday I got my exhaust replaced, started the truck, took it for a drive, stopped at the gas station and it wouldn't restart. It just didn't seem to have the power to turn it over. The gas station is around the corner from my house so I bring over the burb and jump start it. I drive it home, shut it off, still no start. I let it cool down and put a charger on it, no start.

Now it sat a few days, no start after charger on the battery all day. Something really odd though is that when the charger is not on, the headlights won't come on. Turn on the charger, lights work. Did voltage check of battery and its low at about 12 volts. With charger its reading about 13.5 volts, which is about right to draw a charge. But no lights with 12 volts isn't right.

Could it be the solenoid on the starter, or the voltage regulator goofed up or just a weak battery? I haven't done anything but try the charger so far. It's just got me stumped.

HG