If a, b, c are in GP, then show that log an, log bn, log cn are in AP.
To prove: log an, log bn, log cn are in AP.
Given: a, b, c are in GP
Formula used: (i) log ab = log a + log b
As a, b, c are in GP
⇒ b2 = ac
Taking power n on both sides
⇒ b2n = (ac)n
Taking log both side
⇒ logb2n = log(ac)n
⇒ logb2n = log(ancn)
⇒ 2logbn = log(an) + log(cn)
Whenever a,b,c are in AP then 2b = a+c, considering this and the above equation we can say that log an, log bn, log cn are in AP.
Hence Proved