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

case I

When a > 1

let x_{1},x_{2} (0,)

We have, x_{1}<x_{2}

⇒ log_{e} x_{1} < log_{e} x_{2}

⇒ f(x_{1}) < f(x_{2})

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 x_{1}<x_{2}

⇒ log x_{1} < log x_{2}

⇒ []

⇒ f(x_{1}) > f(x_{2})

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

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