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 Powerful Digit (Posted on 2011-03-29)
For a randomly chosen real number x on the interval (0,10) find the exact probability of each:

(1) That x and 2x have the same first digit

(2) That x and x2 have the same first digit

(3) That x2 and 2x have the same first digit.

(4) That x, x2 and 2x all have the same first digit.

First digit refers to the first non-zero digit of the number written in decimal form.

 No Solution Yet Submitted by Jer Rating: 4.0000 (1 votes)

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 Possible solution to part 4. | Comment 5 of 12 |

x, x^2 and 2^x all have the same first digit.

This is true for

(1) cases where the first non-zero digit is 9, with x between x=(log(900))/(log(2)) and x = (3 (log(2)+log(5)))/(log(2))(around 9.81 and 9.96)  and

(2) a lot of small ranges of cases where the first non-zero digit is 1, with x between 0 and 1, more particularly 1/100*(2^(1/2)-1)+1/1000*(2^(1/2)-1)+1/10000*(2^(1/2)-1)+etc...  a series that sums to 0.011111..., or 1/90*(2^(1/2)-1).

Collectively these amount to about 1.52% + 0.46%, an exact formulation being ((8+2^(1/2))log(2)-9log(9/5))/(90log(2)) giving 1.98026822597616101496362510950390% or thereabouts.

Edited on March 30, 2011, 8:05 am
 Posted by broll on 2011-03-30 07:51:56

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