Question
If $$x = {\log_5}3 + {\log _7}5 + {\log _9}7$$ then
A.
$$x \geqslant \frac{3}{2}$$
B.
$$x \geqslant \frac{1}{{\root 3 \of 2 }}$$
C.
$$x \geqslant \frac{3}{{\root 3 \of 2 }}$$
D.
none of these
Answer :
$$x \geqslant \frac{3}{{\root 3 \of 2 }}$$
Solution :
$${\text{AM}} \geqslant {\text{GM}}$$ for positive numbers. So, $$\frac{x}{3} \geqslant \sqrt {{\log_5}3 \cdot {{\log }_7}5 \cdot {{\log }_9}7} = \,\root 3 \of {{{\log }_9}3} $$
$$\therefore \,\,\frac{x}{3} \geqslant \root 3 \of {\frac{1}{{{{\log }_3}9}}} = \root 3 \of {\frac{1}{2}} .$$