Evaluate: ${log}_{7} {log}_{5} {log}_{3} {log}_{2} (2 x^{3} + 5 x^{2} - 14 x)$

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Solution

$f (X) = l o g_{7} l o g_{5} l o g_{3} l o g_{2} (2 x^{3} + 5 x^{2} - 14 x)$
This function is defined only when
$2 x^{3} + 5 x^{2} - 14 x > 0$
$x (2 x^{2} + 5 x - 14) > 0$
$x [2 x^{2} + 5 x - 14] > 0$
$x [(x - 1) (x + \frac{7}{2}] > 0$
This is only possible when
$x \in (- \frac{7}{2}, 0) \cup (1, \infty)$

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