Let's define a notation for "base 1000" where the every "base 1000 digit" consist of a three-base10-digits group.
Thus, the base-10 number 123456789 would read 123 456 789 in base 1000.
As you can see, conversion between these is really, really simple. The reason is that a certain number of digits in the lower base _exactly_ represent a digit in the higher base. This is true for all bases where the higher base is a power of the lower one. For example, base 3 and base 9 could be easily converted, while base 10 and base 20 could not.
For base 2 and base 16, the same applies, only that we decided to write "8" instead of 1000. It's just a different symbol, though...