f(x) = loga x
Let x1, x2∈ (0, ∞) such that x1 < x2.
\(\because\) the function here is a logarithmic function, so either a > 1 or 1 > a > 0.
Case – 1
Let a > 1
x1 < x2
\(\therefore\) logax1 < logax2
\(\therefore\) f(x1) < f(x2)
\(\therefore\) x1 < x2 & f(x1) < f(x2), ∀ x1, x2∈ (0, ∞)
Hence, f(x) is increasing on (0, ∞).
Case – 2
Let, 1 > a > 0
x1 < x2
\(\therefore\) logax1 > logax2
\(\therefore\) f(x1) > f(x2)
\(\therefore\) x1 < x2 & f(x1) > f(x2), ∀ x1, x2∈ (0, ∞)
Thus, for a > 1, f(x) is increasing in its domain.
