How to construct a Turing machine for a language
Try to come up with an algorithm, and then write the simplest possible program in some usual programming language for recognizing L. Then translate that program to a Turing machine.

N is not "in base 10" - it's just a number in your computer's memory, which is probably internally represented in base 2, but your program shouldn't care.

To check if a number contains 1 in base 3 representation, try to build the base-3 representation using the usual base conversion algorithm (digit = x mod 3, proceed with x div 3), and stop when you see a 1.