For a randomly chosen real number x on the interval (0,10) find the exact probability of each:

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

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

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

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

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

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