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Digit-Shifting Constants


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

 b^x=b^nx
(1)

or

 x=n+log_bx.
(2)

The solution is given by

 x=-(W(-b^(-n)lnb))/(lnb),
(3)

where W(x) is the Lambert W-function.

DigitShiftingConstants

The above plot shows log_bx+n-x for b=10 and small values of n. As can be seen, there are two distinct solutions, corresponding to two different branches of W(x) in (3). For n=1, 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,

 10^(0.0102385...)=1.02385...
(4)

and

 10^(2.37581...)=237.581....
(5)

See also

Base, Digit, Logarithm

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Cite this as:

Weisstein, Eric W. "Digit-Shifting Constants." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Digit-ShiftingConstants.html

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