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Algebra Level 4

\[ 2\log_x a + \log_{ax} a + 3\log_b a = 0 \]

For \(a>0 \) and \(b=a^2 x\), solve the equation above.

If the solutions can be expressed as \(a^c\) and \(a^d\), and \(c + d \) can be expressed as \(-\dfrac ef\) for coprime positive integers \(e\) and \(f\), evaluate \(e-f\).

This is part of the set - \(\href{https://brilliant.org/profile/ankit-kumar-c62tb1/sets/my-problems/}{\color{blue} \text{My Problems}}\)

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