CMIC Contest Question

Algebra Level 4

\[ \large \log_{2}x +\log_{8}x+\log_{64}x = \log_{x}2+\log_{x}16+\log_{x}128 \]

Let \(x\) be a real number satisfying the equation above.

If the value of \(\log_{2}x + \log_{x}2\) can be expressed as \(\dfrac{a\sqrt{b}}{c}\) , where \(a\) and \(c\) are coprime positive integers and \(b\) is square-free, compute \(abc.\)

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