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  4. Suppose loga b + logb a = c. The smallest possible integer value of c for all a, b > 1 is

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Q. Suppose $log_a\, b + log_b\, a = c$. The smallest possible integer value of $c$ for all $a, b > 1$ is

KVPYKVPY 2011

Solution:

We have,
$log_a\, b + log_b \,a = c$
$ AM \ge GM$
$\therefore \, \frac{log_a \,b + log_b\,a}{2} \ge \sqrt{log_a\,b\,log_b\,a}$
$\Rightarrow \frac{c}{2} \ge \sqrt{\frac{log\,b}{log\,a} \times \frac{log\,a}{log\,b}}$
$\Rightarrow \frac{c}{2} \ge 1 $
$\Rightarrow c \ge 2$
$\therefore $ Smallest positive integer value of $c = 2$.