question_answer

The value of x, when \[{{\log }_{3}}\text{ }({{\log }_{2}}x)+2\text{ }{{\log }_{9}}({{\log }_{7}}8)=2\], is:

A) 243

B) 27             

C) 343

D) 64

Correct Answer: C

Solution :

Changing the base,

\[\frac{\log (lo{{g}_{2}}x)}{\log 3}+2\frac{\log (lo{{g}_{7}}8)}{\log 9}=2\]

\[\frac{\log (lo{{g}_{2}}x)}{\log 3}+2\frac{\log (lo{{g}_{7}}8)}{2\log 3}=2\]\[\Rightarrow \]\[\log ({{\log }_{2}}x)+\log ({{\log }_{7}}8)=2\log 3\]

\[\log ({{\log }_{2}}x\times {{\log }_{7}}8)=\log 9\Rightarrow {{\log }_{2}}x\times {{\log }_{7}}8=9\]

\[\Rightarrow \frac{3\log x}{\log 7}=9\]

\[\log x=3\log 7\Rightarrow \log x=\log {{7}^{3}}\]\[\Rightarrow \,\,x=343.\]