Prove that the function f(x) = log a x is increasing on (0, ∞) if a 1 and decresing on (0, ∞), if 0

Question

Prove that the function f(x) = log a x is increasing on (0, ∞) if a > 1 and decresing on (0, ∞), if 0 < a < 1.

Philoid · Accepted Answer

case I

When a > 1

let x₁,x₂ (0,)

We have, x₁2

⇒ log_e x₁ < log_e x₂

⇒ f(x₁) < f(x₂)

So, f(x) is increasing in (0,)

case II

When 0 < a < 1

f(x) = log_a x

when a<1 log a < 0

let x₁2

⇒ log x₁ < log x₂

⇒ []

⇒ f(x₁) > f(x₂)

So, f(x) is decreasing in (0,)

Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decresing on (0, ∞), if 0 < a < 1.