Question

`x^(log_5x) > 5` implies ______.

Options

• x ∈ (0, ∞)

• `x ∈ (0, 1/5) ∪ (5, ∞)`

• x ∈ (1, ∞)

• x ∈ (1, 2)

Answer 1

`x^(log_5x) > 5` implies `underlinebb(x ∈ (0, 1/5) ∪ (5, ∞))`.

Explanation:

`x^(log_5x) > 5`

Taking logarithm to base 5 both sides

⇒ log₅x log₅x > 1

⇒ (log₅x)² – 1 > 0

⇒ (log₅x – 1)(log₅x + 1) > 0

∴ log₅x < –1

or log₅x > 1  ...(∵ x > 0)

∴ x < 5^–1 or x > 5

∴ `x ∈ (0, 1/5) ∪ (5, ∞)`