What would the villagers do whenever a log washed ashore?
Could a washed-up log be useful? The answer to this question is somewhat intuitive. People who live on island nations often have limited access to the resources they need to survive. They also have to be careful not to deplete what is available. This includes food, water, shelter, clothing, etc.
This is the case with the villagers in Island of the Blue Dolphins. As Karana mentions, "There are no trees on the island except small ones stunted by the wind." So if one did happen to wash ashore (which according to Karena didn't happen very often), it makes sense that the villagers would put it to good use. They discovered that hollowed-out logs make fine canoes, so whenever one drifts onto the beach, the men typically carry it to the village so a wave can't sweep it back into the ocean.
However, in the beginning of the book, when the Aleut hunting ship arrives, Karana's father instructs some village men to build a canoe from a recent log in the cove and sleep beside it a night. This was so that they could keep watch over the Aleuts and sound an alarm if they tried to sail away without paying the villagers for the otter skins they caught, as per the pact they forged.
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