Test Index

Logarithms

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Question : 28 of 85

Marks: +1, -0

If

0 < a \leq x,

the minimum value of

\log_{a} x + \log_{x} a

is :

1
2
3
5

Solution:

Concept:

Use substitution

t = \log_a x

and apply AM-GM inequality.

Explanation:

Let

t = \log_a x

. Then

\log_x a = \frac{1}{t}

.

Given

0 < a \leq x

, for

a>1

or

0 < a<1

(both positive bases),

t

is positive.

The expression becomes

t + \frac{1}{t}

.

By AM-GM,

t + \frac{1}{t} \geq 2

, with equality when

t=1

i.e.

x=a

.

Hence the minimum value is

2

.

Answer:

2

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