My question is about the booster on my LOA. Why does it only last a certain amount of time? and can I get it to last for 24hrs?
As correctly stated above, Lighthouse of Alexandria, and it's big brother, Royal Albert Hall, work on collections basis.
There is no time frame to act, like using a timer boost.
As far as I know, no Great Building has any powers that work on a timer.
It's per collection, all day, or a limited number of charges like Kraken and the Plunder Ferris Wheel.
Make sure you get St. Marks Great Building too, because that does for coins, what LOA does for supplies.