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If G be the G.M. of the product of K, Set of observations with G.M.’S $ {{G}_{1}},{{G}_{2}},{{G}_{3}},....,{{G}_{K}} $
Respectively, then G equals.
A) $ \log {{G}_{1}}+\log {{G}_{2}}+\log {{G}_{3}}+...+\log {{G}_{k}} $
B) $ {{G}_{1}}{{G}_{2}}{{G}_{3}}...{{G}_{k}} $
C) $ \log {{G}_{1}}.\log {{G}_{2}}.\log {{G}_{3}}...\log {{G}_{k}} $
D) $ k\log \left( {{G}_{1}}.{{G}_{2}}.{{G}_{3}}...{{G}_{k}} \right) $

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Last updated date: 17th Jun 2024
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Answer
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Hint: We will assume $ k $ variables and then find the geometric mean of the variables. For that equation apply logarithmic both sides and use the formula $ \log {{a}^{b}}=b\log a $ and then substitute the product of the $ k $ variables. Now use the formula $ \log \left( ab \right)=\log a+\log b $ to get the result.

Complete step by step answer:
If $ {{x}_{1}},{{x}_{2}},{{x}_{3}},...{{x}_{k}} $ be the $ k $ variables and their product is denoted by $ x={{x}_{1}}.{{x}_{2}}.{{x}_{3}}...{{x}_{k}} $
The geometric mean of the $ k $ variables is $ G={{\left( x \right)}^{\dfrac{1}{k}}} $
Take $ \log $ equation on both sides we have
$\Rightarrow$ $ \log G=\log {{\left( x \right)}^{\dfrac{1}{k}}} $
Using the formula $ \log {{a}^{b}}=b\log a $ in the above equation, then
$\Rightarrow$ $ \log G=\dfrac{1}{k}\log \left( x \right) $
Substitute $ x={{x}_{1}}.{{x}_{2}}.{{x}_{3}}...{{x}_{k}} $ in the above equation we get
$\Rightarrow$ $ \log G=\dfrac{1}{k}\log \left( {{x}_{1}}.{{x}_{2}}.{{x}_{3}}...{{x}_{k}} \right) $
Using the formula $ \log \left( ab \right)=\log a+\log b $ in the above equation, then
 $ \begin{align}
& \log G=\dfrac{1}{k}\log {{x}_{1}}+\dfrac{1}{k}\log {{x}_{2}}+\dfrac{1}{k}\log {{x}_{3}}+...+\dfrac{1}{k}\log {{x}_{k}} \\
 & \log G=\log {{G}_{1}}+\log {{G}_{2}}+\log {{G}_{3}}+...+\log {{G}_{k}} \\
 & \log G=\log \left( {{G}_{1}}{{G}_{2}}{{G}_{3}}...{{G}_{k}} \right) \\
 & G={{G}_{1}}{{G}_{2}}{{G}_{3}}...{{G}_{k}}
\end{align} $

Note:
Please note that we are using the proper logarithmic function at the right place in order to get the result. Some of other logarithmic functions are
 $ \begin{align}
  & \log a-\log b=\log \left( \frac{a}{b} \right) \\
 & \log \left( \frac{1}{y} \right)=\log \left( {{y}^{-1}} \right)=-\log y \\
 & {{\log }_{a}}a=1 \\
 & {{\log }_{a}}\left( {{a}^{b}} \right)=b \\
 & {{a}^{{{\log }_{a}}\left( b \right)}}=b
\end{align} $
Geometric Sequence: In a sequence if the numbers are obtained by multiplying a constant with the previous number (except first number) then that sequence is called a Geometric sequence.
We can write the general form of Geometric Sequence as $ a,ar,a{{r}^{2}},a{{r}^{3}},... $
Where $ a $ is the first term and
      $ r $ is the constant value that is multiplied to the previous term.
Ex: $ 1,2,4,8,16,... $ . Here you can find that each term (except the first term) is obtained by multiplying a constant value $ \left( 2 \right) $ to the previous term. Here we can write $ a=1 $ and $ r=2 $
Geometric Mean: The geometric mean is the special type of average and calculated as $ {{n}^{th}} $ root of the product of $ n $ values. Mathematically geometric mean of the series $ a,{{a}_{1}},{{a}_{2}} $ is
 $ G.M=\sqrt[3]{a\left( {{a}_{1}} \right)\left( {{a}_{2}} \right)} $