If ${log}_{2} x {log}_{3} x {log}_{5} x = {log}_{2} x {log}_{3} x + {log}_{3} x {log}_{5} x + {log}_{5} x {log}_{2} x$ , then value of $x$ which satisfies the given equation is

1

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30

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5

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6

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Solution

The correct options are
A $1$
B $30$
${log}_{2} x {log}_{3} x {log}_{5} x = {log}_{2} x {log}_{3} x + {log}_{3} x {log}_{5} x + {log}_{5} x {log}_{2} x$
Divide both sides by ${log}_{2} x {log}_{3} x {log}_{5} x$
$\Rightarrow 1 = \frac{1}{l o g_{5^{x}}} + \frac{1}{l o g_{2}^{x}} + \frac{1}{l o g_{3}^{x}}$
$\Rightarrow 1 = \frac{1}{l o g x / l o g 5} + \frac{1}{l o g x / l o g 2} + \frac{1}{l o g x / l o g 3}$
$\Rightarrow 1 = \frac{l o g 5}{l o g x} + \frac{l o g 2}{l o g x} + \frac{l o g 3}{l o g x}$
$\Rightarrow 1 = (l o g 5 + l o g 2 + l o g 3) / l o g x$
$\Rightarrow 1 = [l o g (5.2.3)] / l o g x$
$\Rightarrow l o g x = l o g 30$
$x = 30$

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