Log
The log is a small Hebrew liquid measure (about a pint), and it appears only in the Levitical purification ritual for a person cleansed of a skin disease. Four references in Lev 14 fix its place in that ritual: a log of oil is brought along with the lambs and grain offering, waved before Yahweh, poured into the priest's hand, and applied to the cleansed person.
The Log of Oil in the Cleansing Ritual
The initial offering on the eighth day specifies the log of oil alongside the lambs and grain: "And on the eighth day he will take two male lambs without blemish, and one ewe-lamb a year old without blemish, and three tenth parts [of an ephah] of fine flour for a meal-offering, mingled with oil, and one log of oil" (Lev 14:10).
The log of oil is then waved with the trespass-offering: "And the priest will take one of the he-lambs, and offer him for a trespass-offering, and the log of oil, and wave them for a wave-offering before Yahweh" (Lev 14:12).
The priest pours from the log into his own left hand for the application: "And the priest will take of the log of oil, and pour it into the palm of his own left hand" (Lev 14:15).
The poorer-person variant, in which a single lamb stands in place of three, still carries the same log of oil: "and the priest will take the lamb of the trespass-offering, and the log of oil, and the priest will wave them for a wave-offering before Yahweh" (Lev 14:24). Across rich and poor variants, the log measure does not change — the same small unit of oil is the prescribed quantity for cleansing.