If a, b, c are in

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