Not long. We were taught logs, and use of log tables, at Middle School. So probably around about age 11.
I also vaguely recall a couple of lessons where we went over Napier's Bones, and they had us produce equivalents on pieces of paper to cut out and move around.
I believe I still have my school day log tables around somewhere. I'd just have to practice for 1/2 hr to remind myself how to use them. That said, they did have usage instructions in the last few pages.
Having scanned through his book, it makes it seem overly complex.
At middle school, this was taught after having only done simply arithmetic and learning about fractions (rational numbers) in primary school, then decimal fractions in middle school.
The use of logs from the tables was simply a set of basic rules for how to apply them to a few scenarios. I can't recall if it covered trig with those tables, but I doubt it.
I learnt as a child between 10 (when I started middle school), and say around 12 at most. I've forgotten the use, but vaguely recall some way of representing negative numbers in the table (n-bar, with a bar above the digit).
I'm way over 30. I never used log tables after high school, and have forgotten the rules for usage, but recall it didn't take long to learn the first time. *
However for simple uses (multiplication and division) I'd expect I'd be able to pick it up again in at most a weeks worth of practice. It would be made a lot easier now by being able to compare and check calculations with a computer or pocket calculator.
I'd expect any adult able to program a computer to also be able to pick it up in a similar period, or at most a month.
Remember we used to teach this to kids, and expect them to be able to pick it up (if not be accurate in application) in under a weeks worth of lessons.
* Note I didn't even know how to do long multiplication when I learnt, as due to political interference with teaching curriculum, I'd not been taught at primary school.
I also vaguely recall a couple of lessons where we went over Napier's Bones, and they had us produce equivalents on pieces of paper to cut out and move around.
I believe I still have my school day log tables around somewhere. I'd just have to practice for 1/2 hr to remind myself how to use them. That said, they did have usage instructions in the last few pages.