Question

If a and b are positive real numbers other the unity.
then the least value of loga base b + log b base a is:
(11
(c) 2
d) none of these​

Answer 1

Question:

If a and b are positive real numbers other than unity, then the least value of $| log_{b}a + log_{a}b|$ , is

(A)0

(B)1

(C)2

(D)None of these

Solution:

$| log_{b}a + log_{a}b|$

$| log_{b}a + \frac{1}{log_{b}a} |$

$Let \ log_{b}a \ be \ x$

Then,

$| x + \frac{1}{x} |$

According to question we know that,

$x + \frac{1}{x} \geq 2 \ for \ all \ values \ of \ x > 0$

And, $x + \frac{1}{x} \leq -2 \ for \ all \ values \ of \ x < 0$

This happened because we have to find least value other than unity.

Now ,

$| x + \frac{1}{x} | \geq 2 \ for \ all \ values \ of \ x \neq 0$

$| log_{b}a + log_{a}b| \geq 2$

The least value of  $| log_{b}a + log_{a}b|$ is 2.

(C)2 is the correct answer.