The expression [IMAGE] where when simplified is

JEE MainMathematicsModerateLogarithm and its applications

The expression [IMAGE] where $p \geq 2,p \in N;n \in N$ when
simplified is

Show answer and solution

\log_{P}{\log_{P}\sqrt[P]{\sqrt[P]{\sqrt[P]{\ \ldots..\ \sqrt[P]{P}}}}}

Take

\sqrt[P]{\sqrt[P]{\sqrt[P]{\ \ldots..\ \sqrt[P]{P}}}} = \underset{n\ times}{\overset{\left( \left( \left( (P)^{\frac{1}{P}} \right)^{\frac{1}{P}} \right)^{\ldots.} \right)^{\frac{1}{P}}}{︸}}

= (P)^{\frac{1}{P^{n}}}

Now,

\log_{P}^{\log_{P}^{(P)^{\frac{1}{P^{n}}}}}

{{= log}_{P}}^{\frac{1}{P^{n}}^{\log_{P}\ p}}

= \log_{P}\left( \frac{1}{P} \right)^{n}

= n\log_{P}(P)^{- 1}

= - n\log_{P}P

Source: Grass-G / M-05 Mathematics Paper 10 October.docx

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