calculator - What is logarithm & how log table can be constructed by me?

I'm studying properties of logarithm but I don't understand how base e works. Base 10 looks simple while doing calculations of numbers having multiple of 10. As other numbers are not multiple of 10 how one can calculate without using log table. That's why I want know how log table is constructed & how base e works ?
I'm very basic user, I don't get integration, derivatives & summations. Please give answer theoretically or using row concepts. Since I want to construct log table by myself.