Answer :
Question: Prove the following:
[tex]{\sf\: \dfrac{1}{ log_{x}( {x}^{3} {y}^{2} z) } + \dfrac{1}{ log_{xy}( {x}^{3} {y}^{2} z) } + \dfrac{1}{ log_{xyz}( {x}^{3} {y}^{2} z) } = 1 \: } \\ [/tex]
[tex]\large\underline{\sf{Solution-}}[/tex]
Consider,
[tex]\sf\: \dfrac{1}{ log_{x}( {x}^{3} {y}^{2} z) } + \dfrac{1}{ log_{xy}( {x}^{3} {y}^{2} z) } + \dfrac{1}{ log_{xyz}( {x}^{3} {y}^{2} z) } \\ [/tex]
We know,
[tex]\boxed{\sf\: log_{x}(y) = \dfrac{1}{ log_{y}(x) } \: } \\ [/tex]
So, Using this property of logarithms, we get
[tex]\sf\: = \: log_{{x}^{3} {y}^{2} z}({x}) + log_{{x}^{3} {y}^{2} z}({xy}) +log_{{x}^{3} {y}^{2} z}({xyz}) \\ [/tex]
We know,
[tex]\boxed{\sf\: log_{x}(y) \times log_{x}(z) = log_{x}(yz) } \: \\ [/tex]
So, Using this property of logarithms, we get
[tex]\sf\: = \: log_{{x}^{3} {y}^{2} z}({x} \times xy \times xyz) \\ [/tex]
[tex]\sf\: = \: log_{{x}^{3} {y}^{2} z}( {x}^{3} {y}^{2} z ) \\ [/tex]
We know,
[tex]\boxed{\sf\: log_{x}(x) = 1 } \: \\ [/tex]
So, Using this property of logarithms, we get
[tex]\sf\: = \: 1 \\ [/tex]
Hence,
[tex]\implies\sf\:\boxed{\bf\: \dfrac{1}{ log_{x}( {x}^{3} {y}^{2} z) } + \dfrac{1}{ log_{xy}( {x}^{3} {y}^{2} z) } + \dfrac{1}{ log_{xyz}( {x}^{3} {y}^{2} z) } = 1 \: } \\ [/tex]