Question

$f\left(x\right)={\log }_7{\log }_5{\log }_3{\log }_2({2x}^3+{5x}^2-14x)$

Answer 1

For log to be defined ,

(2x³ + 5x² -14x) > 0

x(4 x + 5 - √137)(4x + 5 +√137) > 0

log₂( 2x³ + 5x² -14x) > 0

2x³ + 5x +14x > 1

2x³ + 5x - 14x - 1 > 0________(1)

log₃log₂( 2x³ + 5x² -14x) > 0

2x³ + 5x² - 14x -2 > 0______(2)

log₅log₃log₂( 2x³ + 5x² -14x) >0

2x³ + 5x² -14x - 8 > 0_______(3)

hence,

x(4 x + 5 - √137)(4x + 5 +√137) > 0

and 2x³ + 5x² -14x - 8 > 0 after taking common (1) , (2) and (3)

(x - 2) (x + 4) (2 x + 1)>0

now put in number line then,

domain : $\frac{1}{4}(-5-\sqrt{(}137))<x<0orx>\frac{(-5+\sqrt{(}137))}{4}$