Question

1) Let, a = log₃ log₃ 2. An integer (k) satisfying 1 < 2^{( - k + 3 - a)} < 2, must be less than.
1) 0
2) 1
3) 2
4) 3

Answer 1

Dear student
$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ 1<2^{-\mathrm{k}+3^{-\mathrm{a}}}<2 \\ \Rightarrow 2^0<2^{-\mathrm{k}+3^{-\mathrm{a}}}<2^1 \\ \Rightarrow 0<-\mathrm{k}+3^{-\mathrm{a}}<1 \\ \Rightarrow \mathrm{k}<3^{-\mathrm{a}}<\mathrm{k}+1 \\ \Rightarrow \mathrm{k}<{\left({\log }_{3}2\right)}^{-1}<\mathrm{k}+1 \\ \Rightarrow \mathrm{k}<{\log }_{2}3<\mathrm{k}+1 \\ \Rightarrow 2^{\mathrm{k}}<3<2^{\mathrm{k}+1} \\ \Rightarrow \mathrm{k}=1 \end{gathered}$$
Regards