Solution:
According to the question, a, b and c are in G.P. By making use of the property of Geometric mean, we can write:
$ {{b}^{2}}=~ac $
$ ~{{\left( {{b}^{2}} \right)}^{n}}~=~{{\left( ac \right)}^{n}} $
$ {{b}^{2n}}~=\text{ }{{a}^{n}}~{{c}^{n}} $
Taking log on both the sides we get,
$ log\text{ }{{b}^{2n}}~=\text{ }log\text{ }\left( {{a}^{n}}~{{c}^{n}} \right) $
$ log\text{ }{{\left( {{b}^{n}} \right)}^{2}}~=~log~{{a}^{n}}~+\text{ }log\text{ }{{c}^{n}} $
$ 2\text{ }log\text{ }{{b}^{n}}~=\text{ }log\text{ }{{a}^{n}}~+\text{ }log\text{ }{{c}^{n}} $
$ \therefore \text{ }log\text{ }{{a}^{n}},\text{ }log\text{ }{{b}^{n}},\text{ }log\text{ }{{c}^{n}}~are\text{ }in\text{ }A.P $