You need to understand, that you have to follow specific format, in order to write a good application letter.
Here, you need to know and remember the following important things
1. ***The title of the book as well as the author (It will be better, if you know the author of the name also, for minimising or ending the question)
2. Number of days, you want the book for (Justify your action by mentioning the specific days like from 7th May, 2018 till 15th April, 2018)
Now, check the following letter
Yes, you have to say things in a short and precise manner, since an application letter comes under the category of Formal Letter
I am writing the following letter to the Librarian requesting him to issue book from the book bank
City Montessori School,
Subject (Request to issue a book )
I am X (Your name) student of Class 10th A, requests you to kindly issue X tiled book of Y Author
I can constructively pass my time after school with this informative book.
I will be highly obliged,
Thank You in advance ,
Class 10th Student