f(x) = log_ax

Let x₁, x₂ϵ (0, ∞) such that x₁ < x₂.

the function here is a logarithmic function, so either a > 1 or 1 > a > 0.

Case – 1

Let a > 1

x₁ < x₂

log_ax₁ < log_ax₂

f(x₁) < f(x₂)

x₁ < x₂ & f(x₁) < f(x₂), ∀ x₁, x₂ϵ (0, ∞)

Hence, f(x)is increasing on (0, ∞).

Case – 2

Let, 1 > a > 0

x₁ < x₂

log_ax₁ > log_ax₂

f(x₁) > f(x₂)

x₁ < x₂ & f(x₁) > f(x₂), ∀ x₁, x₂ϵ (0, ∞)

Thus, for a > 1, f(x) is increasing in its domain.