Question

If ${\log }_{16}p=a$ and ${\log }_{64}q=2a$ then

• A. $4p=q$
• B. $p=q$
• C. $p^3=q$
• D. $p^4=q$

Answer 1

Answer: (C)

$p^3=q$

${\log }_{16}p=a,{\log }_{64}q=2a$

$\Rightarrow {\log }_{2^4}p=a,{\log }_{2^6}q=2a$

$\Rightarrow \frac{1}{4}{\log }_{2}p=a,\frac{1}{6}{\log }_{2}q=2a$

$\Rightarrow \frac{1}{2}{\log }_{2}p=2a,\frac{1}{6}{\log }_{2}q=2a$

$\Rightarrow \frac{1}{2}{\log }_{2}p=\frac{1}{6}{\log }_{2}q=2a$

$\Rightarrow \frac{1}{2}{\log }_{2}p=\frac{1}{6}{\log }_{2}q$

$\Rightarrow {\log }_{2}p=\frac{1}{3}{\log }_{2}q$

$\Rightarrow 3{\log }_{2}p={\log }_{2}q$

$\Rightarrow {\log }_2{p}^3={\log }_{2}q$

$\therefore p^3=q$