Are there any two distinct values a and b, whose decimal expansion only contains 0s and 1s, for which tan(a)=b?

Could you explain decimal expansion? l'm not familiar with the term.

Decimal expansion: base 10 writing of a number, the usual way in which we write a number.

So... logarithms? 

No.Logarithms do not matter here. Decimal expansion is just normal writing, such as 10 for ten or 3.1415926... for π 

ONE example comes quickly to mind      tan(0) = 0     but it does not have any 1's