It's continuous. You don't have to reactivate it. It would be 12k if I was making a use-activated item.
Under "Creating Magic items":
"Use-activated or continuous, Spell level × caster level × 2,000 gp, Lantern of revealing"
"This type of item simply has to be used in order to activate it. A character has to drink a potion, swing a sword, interpose a shield to deflect a blow in combat, look through a lens, sprinkle dust, wear a ring, or don a hat. Use activation is generally straightforward and self-explanatory.
Many use-activated items are objects that a character wears. Continually functioning items are practically always items that one wears. A few must simply be in the character’s possession (on his person). However, some items made for wearing must still be activated. Although this activation sometimes requires a command word, usually it means mentally willing the activation to happen. The description of an item states whether a command word is needed in such a case."
Calypso