For a real number x, if 12, log3(2x-9)log34, and log5(2x+172)log54 are in arithmetic progression, then the common difference is
Explanation:
Given, 12, log3(2x-9)log34 and log5(2x+172)log54 are in arithmetic progression,
⇒ 1/2, log4 (2x - 9) and log4 (2x + 17/2) are in AP
⇒ 2 × log4 (2x - 9) = 1/2 + log4 (2x + 17/2)
⇒ log4 (2x - 9)2 = log4 2 + log4 (2x + 17/2)
⇒ log4 (2x - 9)2 = log4 2 × (2x + 17/2)
⇒ (2x - 9)2 = 2 × (2x + 17/2)
⇒ (a - 9)2 = 2a + 17 [Assuming 2x = a]
⇒ a2 - 18a + 81 = 2a + 17
⇒ a2 - 20a + 64 = 0
⇒ (a - 16)(a - 4) = 0
⇒ a = 2x = 16 [4 is rejected as (a - 9) cannot be negative]
Now the first term of the AP = 1/2 and
second term of the AP = log4 (2x - 9) = log4 (16 - 9) = log4 7
Common difference = log4 7 - 1/2 = log4 7 - log4 2 = log4 7/2
Hence, option (a).
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