If m is the geometric mean of \({\left( {\frac{{\rm{y}}}{{\rm{z}}}} \right)^{\log \left( {{\rm{yz}}} \right)}},{\rm{\;}}{\left( {\frac{{\rm{z}}}{{\rm{x}}}} \right)^{\log \left( {{\rm{zx}}} \right)}}{\rm{\;and\;}}{\left( {\frac{{\rm{x}}}{{\rm{y}}}} \right)^{\log \left( {{\rm{xy}}} \right)}}\) then what is the value of m?
1
Concept:
- Geometric mean = \({\left( {{{\rm{a}}_1} \times {{\rm{a}}_2} \times \ldots . \times {{\rm{a}}_{\rm{n}}}} \right)^{\left( {\frac{1}{{\rm{n}}}} \right)}}{\rm{\;}}\)
Where n is the number of terms.
- log (xy) = log (x) + log (y)
- log(x/y) = log x – log y
- \(\log \left( {{{\rm{a}}^{\rm{n}}}} \right) = {\rm{n\;log}}\left( {\rm{a}} \right)\)
- ln a = b ⇒ a = eb
- \({{\rm{a}}^{\rm{m}}} \times {{\rm{a}}^{\rm{n}}} = {{\rm{a}}^{{\rm{m}} + {\rm{n}}}}\)
- \({\rm{\;}}\frac{{{{\rm{a}}^{\rm{m}}}}}{{{{\rm{a}}^{\rm{n}}}}} = {{\rm{a}}^{{\rm{m}} - {\rm{n}}}}\)
Calculation:
Given terms:
\({\left( {\frac{{\rm{y}}}{{\rm{z}}}} \right)^{\log \left( {{\rm{yz}}} \right)}},{\rm{\;}}{\left( {\frac{{\rm{z}}}{{\rm{x}}}} \right)^{\log \left( {{\rm{zx}}} \right)}}{\rm{\;and\;}}{\left( {\frac{{\rm{x}}}{{\rm{y}}}} \right)^{\log \left( {{\rm{xy}}} \right)}}\)
Here, the number of terms = n = 3
\({\rm{Let}},{\rm{\;}}{{\rm{a}}_1} = {\left( {\frac{{\rm{y}}}{{\rm{z}}}} \right)^{\log \left( {{\rm{yz}}} \right)}},{\rm{\;}}{{\rm{a}}_2} = {\rm{\;}}{\left( {\frac{{\rm{z}}}{{\rm{x}}}} \right)^{\log \left( {{\rm{zx}}} \right)}}{\rm{\;and\;}}{{\rm{a}}_3} = {\left( {\frac{{\rm{x}}}{{\rm{y}}}} \right)^{\log \left( {{\rm{xy}}} \right)}}\)
\({\rm{Geometric\;mean}} = {\rm{m}} = {\left( {{{\rm{a}}_1} \times {{\rm{a}}_2} \times {{\rm{a}}_3}} \right)^{\frac{1}{3}}}\)
Now,
\({{\rm{a}}_1} \times {{\rm{a}}_2} \times {{\rm{a}}_3} = {\left( {\frac{{\rm{y}}}{{\rm{z}}}} \right)^{\log \left( {\rm{y}} \right) + \log ({\rm{z}})}} \times {\rm{\;}}{\left( {\frac{{\rm{z}}}{{\rm{x}}}} \right)^{\log \left( {\rm{z}} \right) + \log ({\rm{x}})}} \times {\rm{\;}}{\left( {\frac{{\rm{x}}}{{\rm{y}}}} \right)^{\log \left( {\rm{x}} \right) + \log \left( {\rm{y}} \right)}}\)
\(\Rightarrow \frac{{{{\rm{y}}^{\log {\rm{y}}}} \times {{\rm{y}}^{\log {\rm{z}}}}}}{{{{\rm{z}}^{\log {\rm{y}}}} \times {{\rm{z}}^{\log {\rm{z}}}}}} \times \frac{{{{\rm{z}}^{\log {\rm{z}}}} \times {{\rm{z}}^{\log {\rm{x}}}}}}{{{{\rm{x}}^{\log {\rm{z}}}} \times {{\rm{x}}^{\log {\rm{x}}}}}} \times \frac{{{{\rm{x}}^{\log {\rm{x}}}} \times {{\rm{x}}^{\log {\rm{y}}}}}}{{{{\rm{y}}^{\log {\rm{x}}}} \times {{\rm{y}}^{\log {\rm{y}}}}}}\) (\(\because {{\rm{a}}^{{\rm{m}} + {\rm{n}}}} = {{\rm{a}}^{\rm{m}}} \times {{\rm{a}}^{\rm{n}}}\))
\(\Rightarrow \frac{{{{\rm{y}}^{\log {\rm{z}}}}}}{{{{\rm{z}}^{\log {\rm{y}}}}}} \times \frac{{{{\rm{z}}^{\log {\rm{x}}}}}}{{{{\rm{x}}^{\log {\rm{z}}}}}} \times \frac{{{{\rm{x}}^{\log {\rm{y}}}}}}{{{{\rm{y}}^{\log {\rm{x}}}}}}\)
\(\Rightarrow {{\rm{a}}_1} \times {{\rm{a}}_2} \times {{\rm{a}}_3} = {\left( {\frac{{\rm{y}}}{{\rm{x}}}} \right)^{\log {\rm{z}}}} \times {\left( {\frac{{\rm{x}}}{{\rm{z}}}} \right)^{\log {\rm{y}}}} \times {\left( {\frac{{\rm{z}}}{{\rm{y}}}} \right)^{\log {\rm{x}}}}\)
Taking log on both sides,
\(\log \left( {{{\rm{a}}_1} \times {{\rm{a}}_2} \times {{\rm{a}}_3}} \right) = \log {\rm{z}}\log \left( {\frac{{\rm{y}}}{{\rm{x}}}} \right) \times \log {\rm{y}}\log \left( {\frac{{\rm{x}}}{{\rm{z}}}} \right) \times \log {\rm{x}}\log \left( {\frac{{\rm{z}}}{{\rm{y}}}} \right){\rm{\;}}\)
\(= \left( {\log {\rm{z}}\log {\rm{y}} - \log {\rm{z}}\log {\rm{x}}} \right) + \left( {\log {\rm{y}}\log {\rm{x}} - \log {\rm{y}}\log {\rm{z}}} \right) + \left( {\log {\rm{x}}\log {\rm{z}} - \log {\rm{x}}\log {\rm{y}}} \right) = 0\)
\(\Rightarrow {{\rm{a}}_1} \times {{\rm{a}}_2} \times {{\rm{a}}_3} = {{\rm{e}}^0}\) (\(∵ ln a = b ⇒ a = e\))
∴ a1 × a2 × a3 = 1
Now, geometric mean = \({\rm{m}} = {\left( {{{\rm{a}}_1} \times {{\rm{a}}_2} \times {{\rm{a}}_3}} \right)^{\frac{1}{3}}}\)
∴ m = 1
Thus, option (3) is correct.
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