If $l o g_{6} 9 - l o g_{9} 27 + l o g_{8} x = l o g_{64} x - l o g_{6} 4$ then the value of x is :

1/2

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1/4

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1/8

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1/6

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Solution

The correct option is C 1/8
$l o g_{6}^{9} - l o g_{9}^{27} + l o g_{8}^{x} = l o g_{64}^{x} - l o g_{6}^{4}$
$l o g_{6}^{9} + l o g_{6}^{4} - l o g_{9}^{27} = l o g_{64}^{x} - l o g_{8}^{x}$
$\Rightarrow l o g_{6}^{9 \times 4} - l o g_{3^{2}}^{3^{3}} = l o g_{8^{2}}^{x} - l o g_{8}^{x}$
$\Rightarrow l o g_{6}^{6^{2}} - \frac{3}{2} l o g_{3}^{3} = \frac{1}{2} l o g_{8}^{x} - l o g_{8}^{x}$
$\Rightarrow 2 - \frac{3}{2} = - \frac{1}{2} l o g_{8}^{x}$
$\Rightarrow \frac{1}{2} = \frac{- 1}{2} l o g_{8}^{x}$
$\Rightarrow l o g_{8}^{x} = - 1$
$\Rightarrow x = 8^{- 1}$
$∴ x = \frac{1}{8}$ .

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