Logarithm System

Algebra Level 2

{logxw=24logyw=40logxyzw=12\Large\begin{cases} \log_x w & = & 24 \\ \log_y w & = & 40 \\ \log_{xyz} w & = & 12 \end{cases}

If x,y,zx,y,z are real numbers greater than 1 and ww is a positive number satisfying the system above, then find the value of (logwz)1\left (\log_w z \right)^{-1}.

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