Digit-Shifting Constants

Given a real number , find the powers of a base  that will shift the digits of  a number of places  to the left. This is equivalent to solving

(1)

or

(2)

The solution is given by

(3)

where  is the Lambert W-function.

The above plot shows  for  and small values of . As can be seen, there are two distinct solutions, corresponding to two different branches of  in (3). For , 2, ..., these solutions are approximately given by 0.137129, 0.0102386, 0.00100231, 0.000100023, 0.0000100002, ..., and 1, 2.37581, 3.55026, 4.66925, 5.76046, ..., respectively. For example,

(4)

and

(5)