Question

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

• A. ${\log }_{30}15$
• B. ${\log }_{30}45$
• C. ${\log }_{900}15-{\log }_{900}45$
• D. ${\log }_{30}16-{\log }_{30}2$

Answer 1

Answer: (A), (B), (C)

${\log }_{900}15-{\log }_{900}45$
${\log }_{30}15$
${\log }_{30}45$

$x={\log }_{30}3$ and

$y={\log }_{30}5$

Now,

${\log }_{30}15={\log }_{30}3+{\log }_{30}5=x+y$

${\log }_{30}45={\log }_{30}{3}^2+{\log }_{30}5=2x+y$

${\log }_{{30}^2}15-{\log }_{{30}^2}45=\frac{1}{2}({\log }_{30}3+{\log }_{30}5)-\frac{1}{2}({\log }_{30}5+{\log }_{30}{3}^2)=\frac{1}{2}(x+y)-\frac{1}{2}(2x+y)=-\frac{1}{2}x$