Question

If $\frac{1}{3}{\log }_{3}M+3{\log }_{3}N=1+{\log }_{0.008}5$ then

• A. $M^9=\frac{9}{N}$
• B. $N^9=\frac{9}{M}$
• C. $M^3=\frac{3}{N}$
• D. $N^9=\frac{3}{M}$

Answer 1

The correct option is B $N^9=\frac{9}{M}$

$\frac{1}{3}{\log }_{3}M+3{\log }_{3}N=1+{\log }_{0.008}5$
$\frac{1}{3}{\log }_{3}M+3{\log }_{3}N=1+\frac{\log 5}{\log 0.008}$
$\frac{1}{3}{\log }_{3}M+3{\log }_{3}N=1+\frac{\log 5}{\log \frac{8}{1000}}$
$\frac{1}{3}{\log }_{3}M+3{\log }_{3}N=1+\frac{\log 5}{\log 8-\log 1000}$
$\frac{1}{3}{\log }_{3}M+3{\log }_{3}N=1+\frac{\log 5}{3(\log 2-\log 10)}$
$\frac{1}{3}{\log }_{3}M+3{\log }_{3}N=1+\frac{\log 5}{3(\log \frac{1}{5}}$
${\log }_3{M}^{\frac{1}{3}}{N}^3=1-\frac{1}{3}$
$\log M^{\frac{1}{3}}{N}^3=\frac{2}{3}\log 3$
$M^{\frac{1}{3}}{N}^3=3^{\frac{2}{3}}$
$M{N}^9=9$
$N^9=\frac{9}{M}$