Question

If ${\log }_{ab}a=\frac{1}{3}$, then find the value of ${\log }_{ab}\left(\frac{\sqrt{a}}{\sqrt[3]{3b}}\right)$

• A. $-18$
• B. $18$
• C. $\frac{1}{18}$
• D. $\frac{-1}{18}$

Answer 1

Answer: (D)

$\frac{-1}{18}$

Given ${\log }_{ab}a=\frac{1}{3}\Rightarrow 3{\log }_{ab}a=1$

$\Rightarrow {\log }_{ab}{a}^3=1\Rightarrow a^3=ab\Rightarrow a^2=b$,.....(1) as $a>0$

Therefore

${\log }_{ab}\left(\frac{\sqrt{a}}{\sqrt[3]{3b}}\right)=\frac{1}{6}{\log }_{ab}{\left(\frac{\sqrt{a}}{\sqrt[3]{3b}}\right)}^6=\frac{1}{6}{\log }_{ab}\frac{a^3}{b^2}$

$=\frac{1}{6}{\log }_{a^3}\frac{a^3}{a^4}=\frac{1}{6}{\log }_{a^3}{a}^{-1}$ using (1)

$=\frac{1}{6}\cdot \frac{1}{3}(-1){\log }_{a}a=-\frac{1}{18}$