Let `a=(log)_3(log)_3 2.` An integer `k` satisfying `1<2^(-k+3^((-a)))<2,` must be less than ..........

Options

A. 1
B. 2
C. 0
D. `-1`

Answer

A

Explanation

We have,
`a = "log"_(3)("log"_(3) 2) rArr 3^(a) = "log"_(3) 2 rArr "log"_(2) 3 = 3^(-a)`
Now, `1 lt 2^((-k+3^(-a))) lt 2`
`rArr 0 lt -k + 3^(-a) lt 1`
`rArr 0 lt -k + "log"_(2) 3 lt 1`
`rArr k lt "log"_(2) 3 lt k +1`
Now, `"log"_(2) 3 gt 1 rArr k +1 gt 1 rArr k gt0`
and, `"log"_(2) 3 lt 2 rArr k lt 2`
`therefore 0 lt k lt 2 rArr k = 1 " " [therefore "k is an integer"]`