Q.

If \( \frac = \frac = \frac \), find the value of k.

A. 8/3

B. 10/3

C. 3

D. \( \sqrt \)

Find the relation between 4, 6 & 9 and then using it, the relation between \(\log_4, \; \log_6, \; \log_9\) and then replace them with terms related to 2, k & n.

A. 8/3

Approach 1:

4, 6, 9 are in GP

⇒ \(\log_4, \; \log_6, \; \log_9 \) are in AP

⇒ \(\log_x, \; \log_x, \; \log_x \) are in HP

⇒ 2n, kn, 4n are in HP

⇒ 2, k, 4 are in HP

⇒ \( k = \frac    \)

Approach 2:

If the given ratios is equal to n, then

\(\log_x=2n \implies \log_4=\frac\)

Similarly, \(\log_6=\frac\) and \(\log_9 =\frac\)

Next, 62 = 4 × 9 \(\implies 2\log_6 = \log_4 + \log_9\)

i.e. \(\frac = \frac + \frac \)

The n gets cancelled from all terms and you should be able to solve it to get value of k.