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)
|
