Prove that the function is strictly increasing on ]0, ∞[ when a 1 and strictly decreasing on ]0, ∞[ when 0

Question

Prove that the function is strictly increasing on ]0, ∞[ when a > 1 and strictly decreasing on ]0, ∞[ when 0 < a < 1.

Philoid · Accepted Answer

Consider ƒ(x)=log_a x

domain of f(x) is x>0

⇒ for a>1, ln(a)>0,

hence f’(x) >0 which means strictly increasing.

⇒ for 0

hence f’(x)<0 which means strictly decreasing.