What part of the country are you in? Your town historian would be a good place to start; it depends on what else you know about the people associated with the ships log. Was the captain or first mate family?

For now, storing it is best done by either keeping it in. Bookshelf (if bound) or inside a Manila folder if (loose sheets.)

I would also recommend getting in contact with the historical society for whatever port it called home.

As for monetary value that depends entirely on who's ship log it was during the War of 1812. I'm not an appraiser though.