Logarithms!

Algebra Level 4

\[\large \begin{eqnarray} \log_2 x + \log_4 y + \log_4 z &=& 2 \\ \log_3 y + \log_9 z + \log_9 x &=& 2 \\ \log_4 z + \log_{16} x + \log_{16} y &=& 2 \end{eqnarray} \]

Solve the system equations where \(x\) , \(y\) and \(z\) are positive real numbers.

If the value \(x,y,z\) can be written in the form \(\dfrac{a_x}{b_x} , \dfrac{a_y}{b_y} , \dfrac{a_z}{b_z}\) for co-prime \(a_i , a_j\), find the value of \(a_x+b_x+a_y+b_y+a_z+b_z\).

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