f(x) = log7(log5(log3(log2(2x3 + 5x2 - 14x))))
To defined function f(x), we have
log5(log3(log2(2x3 + 5x2 - 14x))) > 0
⇒ log3(log2(2x3 + 5x2 - 14x)) >5° or 1
⇒ log2(2x3 + 5x2 - 14x) > 31 or 3 ( ∵ log is increasing function)
⇒ 2x3 + 5x2 - 14x >23 or 8
⇒ 2x3 + 5x2 - 14x - 8 > 0
⇒ (x - 2) (2x2 + 9x + 4) > 0
⇒ (x - 2) (x + 4) (2x + 1) > 0
Domain of given function f is (-4, -1/2) U (2, \(\infty\))