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Question

Prove that: $$\displaystyle\log_7 \log _{7}\sqrt{7\sqrt{\left ( 7\sqrt{7} \right )}}=1-3\log _{7}2$$


Solution

Let $$L=\displaystyle \log _{ 7 }{ \log _{ 7 }{ \sqrt { 7\left( \sqrt { 7\sqrt { 7 }  }  \right)  }  }  }$$

$$=\log _{ 7 }{ \log _{ 7 }{ \sqrt { 7\left( \sqrt { 7^{ 3/2 } }  \right)  }  }  } $$

$$=\log _{ 7 }{ \log _{ 7 }{ \sqrt { 7\left( 7^{ 3/4 } \right)  }  }  } $$

$$=\log _{ 7 }{ \log _{ 7 }{ \sqrt { 7^{ 7/4 } }  }  } $$

$$=\log _{ 7 }{ \log _{ 7 }{ 7^{ 7/8 } }  } $$

$$=\log _{ 7 } \left( \frac { 7 }{ 8 } \log _{ 7 } 7 \right) $$

$$ =\log _{ 7 }{ \left( \dfrac { 7 }{ 8 }  \right)  }, $$           ($$\because \log_a a=1$$)

$$=\log _{ 7 }{ 7 } -\log _{ 7 }{ 8 }$$

$$ =1-\log _{ 7 }{ { 2 }^{ 3 } }, $$          ($$\because \log_a a=1$$)

$$L=1-3\log _{ 7 }{ 2 } $$

$$\displaystyle \log _{ 7 }{ \log _{ 7 }{ \sqrt { 7\left( \sqrt { 7\sqrt { 7 }  }  \right)  }  }  }=1-3\log _{ 7 }{ 2 } $$

Hence, proved

Mathematics

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