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Logarithms

If 3log37x=...

Question

If $3^{(l o g_{3} 7)^{x}} = 7^{(l o g_{7} 3)^{x}}$ then the value of x will be:

A

\frac{1}{2}

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B

\frac{1}{4}

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C

\frac{1}{8}

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D

\frac{1}{12}

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Solution

The correct option is A $\frac{1}{2}$
$3^{(l o g_{3}^{7})^{x}} = 7^{(l o g_{7}^{3})^{x}}$
Take log on both sides
$l o g^{3^{(l o g_{3}^{7})^{x}}} = l o g^{7^{(l o g_{7}^{3})^{x}}}$
$(l o g_{3}^{7})^{x} l o g 3 = (l o g_{7}^{3})^{x} l o g^{7}$
$\frac{(l o g_{3}^{7})^{x}}{(l o g_{7}^{3})^{x}} = \frac{l o g 7}{l o g 3}$
$l o g_{3}^{7} = \frac{1}{l o g_{7}^{3}}$
$\Rightarrow (l o g_{3}^{7})^{x} \cdot (l o g_{3}^{7})^{x} = l o g_{3}^{7}$
$\Rightarrow (l o g_{3}^{7})^{2 x} = l o g_{3}^{7}$
$\Rightarrow 2 x = 1$
$\Rightarrow x = \frac{1}{2}$ .

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