Question

If $|x-{\log }_{\sqrt{3}/2}\left(\frac{64}{27}\right)|=1$, then the value of $x$ is/are

• A. $5$
• B. $7$
• C. $-7$
• D. $-5$

Answer 1

Answer: (C), (D)

The correct options are
C $-7$
D $-5$

Given:
$|x-{\log }_{\sqrt{3}/2}\left(\frac{64}{27}\right)|=1$
Assuming
${\log }_{\sqrt{3}/2}\left(\frac{64}{27}\right)=a\Rightarrow \frac{64}{27}={\left(\sqrt{3}/2\right)}^a\Rightarrow \frac{2^6}{3^3}={\left(\frac{\sqrt{3}}{2}\right)}^a\Rightarrow \frac{2^6}{(\sqrt{3}{)}^6}={\left(\frac{\sqrt{3}}{2}\right)}^a\Rightarrow {\left(\frac{2}{\sqrt{3}}\right)}^6={\left(\frac{\sqrt{3}}{2}\right)}^a\Rightarrow a=-6$
Now,
$|x-(-6)|=1\Rightarrow |x+6|=1\Rightarrow x+6=\pm 1\Rightarrow x=-6\pm 1\Rightarrow x=-7,-5$