Solve the system of equations:
$l o g_{a} x l o g_{a} (x y z) = 48 l o g_{a} y l o g_{a} (x y z) = 12, a > 0, a \neq 1 l o g_{a} z l o g_{a} (x y z) = 84$

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Solution

$l o g_{a} x l o g_{a} (x y z) = 48$ ..... (1)
$l o g_{a} y l o g_{a} (x y z) = 12$ ...... (2)
$l o g_{a} z l o g_{a} (x y z) = 84$ ..... (3)
Adding (1), (2) & (3)
$(l o g_{a} x + l o g_{a} y + l o g_{a} z) l o g_{a} (x y z) = 144 {l o g_{a} (x y z)}^{2} = 144 l o g_{a} (x y z) = \pm 12 l o g_{a} x l o g_{a} (x y z) = 48 l o g_{a} x \times 12 = 48 x = a^{4}$
Similarly y = a
$z = a^{7}$
Or
$x = a^{- 4} y = a^{- 1} z = a^{- 7}$

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{log}_{a} x {log}_{a} (x y z) = 48

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