Let `x` be chosen at random from the interval `(0,1)`.The probability that `[log_(10)4x]=[log_(10)x]` is `k//12` ([.] denotes G.I.F), then `k` is

Answer

2

Explanation

Let `[log_(10)4x]=k=[log_(10)x]` then `xepsilon[10^(x)(10^(k+1))/4], `for `kgt0` & `k=0 (10^(k), (10^(k+1))/4)` is beyond `(0,1)` but for `klt0` the whole interval is in `(0,1)`.
So, favouable `x` is obtained for `k=-1, -2, -3`………..
for `k=-1, xepsilon [1/10, 1/4)`
for `k=-2, xepsilon[1/100,1/40)`..........
sum of length of these intervals
`=(1/4-1/10)+(1/40-1/100)+.......`
`=3/20(1+ 1/10+1/100+...)=1/6`
Required probability `=(1/6)/(1-0)=1/6`