Question

$\text{Let}a={\log }_3{\log }_{3}2\text{. An integer}k\text{satisfying}1<2^{(-k+3^{-a})}<2\text{, must be less than}$

• A. one
• B. 1
• C. 1.00
• D. ONE
• E. 1.0
• F. 01
• G. One

Answer 1

Answer: (A), (B), (C), (D), (E), (F), (G)

$1<2^{(-k+3^{-a})}<2\Rightarrow 0<-k+3^{-a}<1$
Given, $a={\log }_3{\log }_{3}2$
$\Rightarrow 3^a={\log }_{3}2\Rightarrow 3^{-a}={\log }_{2}3$...(i)
$\therefore k<{\log }_{2}3<2$ $\dots \left(ii\right)$
and $1+k>{\log }_{2}3>1\Rightarrow k>0$ $\dots \left(iii\right)$
From Eqs. (ii) and (iii),
$0<k<2$
$k=1\text{[since,}k\text{is an integer]}$