Question:
If $a, b, c$ are in GP, then show that $\log a^{n}, \log b^{n}, \log c^{n}$ are in AP.
Solution:
To prove: $\log a^{n}, \log b^{n}, \log c^{n}$ 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
$\Rightarrow \mathrm{b}^{2}=\mathrm{ac}$
Taking power n on both sides
$\Rightarrow \mathrm{b}^{2 \mathrm{n}}=(\mathrm{ac})^{\mathrm{n}}$
Taking log both side
$\Rightarrow \log b^{2 n}=\log (a c)^{n}$
$\Rightarrow \log b^{2 n}=\log \left(a^{n} c^{n}\right)$
$\Rightarrow 2 \log b^{n}=\log \left(a^{n}\right)+\log \left(c^{n}\right)$
Whenever $a, b, c$ are in $A P$ then $2 b=a+c$, considering this and the above equation we can say that $\log a^{n}, \log b^{n}, \log c^{n}$ are in $A P$.
Hence Proved