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 x^{2} have the same first digit
(3) That x^{2} and 2^{x} have the same first digit.
(4) That x, x^{2} and 2^{x} all have the same first digit.
First digit refers to the first nonzero 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 nonzero 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 nonzero 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 20110330 07:51:56 