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 $abc.$