Sol. Note that the domain of f(x) is the set of all positive real numbers i.e. f(x) is defined for all x > 0 and x ≠ 1.

Now,

f(x) =

⇒ f ′ (x) =

For f(x) to be increasing, we must have

f ′ (x) > 0 ⇒ > 0

⇒ log x – 1 > 0 [  (log x) 2 > 0 for x > 0]

⇒ log x > 1

⇒ x > e 1 [  log a x > N ⇒ x > a N for a > 1]

⇒ x ∈ (e, ∞ )

So, f(x) is increasing on (e, ∞ )

For f(x) to be decreasing, we must have

f ′ (x) < 0 ⇒ < 0

⇒ log x – 1 < 0 [  (log x) 2 > 0 for x > 0]

⇒ log x < 1

⇒ x < e 1 ⇒ x ∈ (0, e) [  f(x) is defined for x > 0]

So, f(x) is decreasing on (0, 1) ∪ (1, e)