Problem: If 7(3(2x))=0\log _{7}\left(\log _{3}\left(\log _{2} x\right)\right)=0log7(log3(log2x))=0, then x−1/2x^{-1 / 2}x−1/2 equals
Answer Choices:
A. 13\dfrac{1}{3}31
B. 123\dfrac{1}{2 \sqrt{3}}231
C. 133\dfrac{1}{3 \sqrt{3}}331
D. 142\dfrac{1}{\sqrt{42}}421
E. none of these
Solution:
7(3(2x))=0⇒3(2x)=1⇒2x=3⇒x=23=8\begin{aligned} \log _{7}\left(\log _{3}\left(\log _{2} x\right)\right)=0 &\Rightarrow \log _{3}\left(\log _{2} x\right)=1 \\ &\Rightarrow \log _{2} x=3 \Rightarrow x=2^{3}=8 \end{aligned} log7(log3(log2x))=0⇒log3(log2x)=1⇒log2x=3⇒x=23=8
So x−1/2=18=122x^{-1 / 2}=\dfrac{1}{\sqrt{8}}=\dfrac{1}{2 \sqrt{2}}x−1/2=81=221.