Answer

Verified

402.9k+ views

**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)} $

Recently Updated Pages

The base of a right prism is a pentagon whose sides class 10 maths CBSE

A die is thrown Find the probability that the number class 10 maths CBSE

A mans age is six times the age of his son In six years class 10 maths CBSE

A started a business with Rs 21000 and is joined afterwards class 10 maths CBSE

Aasifbhai bought a refrigerator at Rs 10000 After some class 10 maths CBSE

Give a brief history of the mathematician Pythagoras class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Name 10 Living and Non living things class 9 biology CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Write the 6 fundamental rights of India and explain in detail