Why is it necessary to use binary numbers in logspace?

Question

I have noticed that a lot of problems that are in L and NL use binary numbers. I don't understand why this is the case. Does a TM use less space by storing a binary number, than a "normal" one. In my head, it uses less space to store the number 0, than the binary ecquivalent number 0000. Hope someone can help me understand :)