[Smaug123]

This is false. As soon as you get to first-year undergraduate maths, one discards log-to-base-10, all but permanently. Nearly all the rest of integer arithmetic is performed in an arbitrary base (if it's being considered "additively" in some sense), or much more commonly, using prime factorisation, which is independent of any base.

Areas of maths which are not number theory basically never mention the number which is twice five at all, and could be done happily without ever knowing that there was some privileged base in which we count the naturals.