Question

If $\log (\sqrt{54}\times \sqrt[3]{3243})$ can be expressed as $\frac{1}{2}\log 2+\frac{m}{6}\log 3$, then value of $m$ will be

• A. $18$
• B. $17$
• C. $19$
• D. $12$

Answer 1

Answer: (C)

$19$

$\log (\sqrt{54}\times \sqrt[3]{3243})=\log (\sqrt{2\times 3^3}\times \sqrt[3]{33^5})=\log (2^{\frac{1}{2}}\cdot 3^{\frac{3}{2}}\cdot 3^{\frac{5}{3}})=\log (2^{\frac{1}{2}}\cdot 3^{(\frac{3}{2}+\frac{5}{3})})=\log (2^{\frac{1}{2}}+\log 3^{\frac{19}{6}})=\frac{1}{2}\log 2+\frac{19}{6}\log 3$