Q. 35.0( 1 Vote )

# Write the set of values of ‘a’ for which f(x) = log_{a} x is increasing in its domain.

Answer :

f(x) = log_{a}x

Let x_{1}, x_{2}ϵ (0, ∞) such that x_{1} < x_{2}.

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

Case – 1

Let a > 1

x_{1} < x_{2}

log_{a}x_{1} < log_{a}x_{2}

f(x_{1}) < f(x_{2})

x_{1} < x_{2} & f(x_{1}) < f(x_{2}), ∀ x_{1}, x_{2}ϵ (0, ∞)

Hence, f(x)is increasing on (0, ∞).

Case – 2

Let, 1 > a > 0

x_{1} < x_{2}

log_{a}x_{1} > log_{a}x_{2}

f(x_{1}) > f(x_{2})

x_{1} < x_{2} & f(x_{1}) > f(x_{2}), ∀ x_{1}, x_{2}ϵ (0, ∞)

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

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