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If log 16p=...

Question

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

A

4 p = q

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B

p = q

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C

p^{3} = q

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D

p^{4} = q

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Solution

The correct option is D $p^{3} = q$
${log}_{16} p = a, {log}_{64} q = 2 a$

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

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

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

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

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

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

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

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

$∴ p^{3} = q$

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