If a, b, c are in GP, then show that log

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

 

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