Question

If $x={\log }_{30}3$ and $y={\log }_{30}5$, then identify the option(s) which can be represented in terms of $x$ and $y$.

• A. ${\log }_{30}16$
• B. ${\log }_{30}45$
• C. ${\log }_{900}90-{\log }_{900}45$
• D. ${\log }_{30}16-{\log }_{30}2$

Answer 1

Answer: (B)

${\log }_{30}45$

$A.{\log }_{30}16={\log }_{30}{2}^4=4{\log }_{30}2B.{\log }_{30}45={\log }_{30}(3^2\times 5)={\log }_{30}{3}^2+{\log }_{30}5=2{\log }_{30}3+{\log }_{30}5C.{\log }_{900}90-{\log }_{900}45={\log }_{{30}^2}90-{\log }_{{30}^2}45=\frac{1}{2}[{\log }_{30}90-{\log }_{30}45]=\frac{1}{2}[{\log }_{30}\left(3^2.5.2\right)-{\log }_{30}\left(3^2.5\right)]=\frac{1}{2}[{\log }_{30}\left(3^2.5\right)+{\log }_{30}2-{\log }_{30}\left(3^2.5\right)]=\frac{{\log }_{30}2}{2}D.{\log }_{30}16-{\log }_{30}2=4{\log }_{30}2-{\log }_{30}2=3{\log }_{30}2$

Hence, only option B can be expressed in terms of $x$ and $y$.