Question

Determine o valor da expressão, com n sendo inteiro:

$- log_{n} [log_{n} (\sqrt[n]{\sqrt[n]{\sqrt[n]{\sqrt[n]{\sqrt[n]{n} } } } } ) ]$

Answer 1

$\displaystyle -\text{log}_{\displaystyle \ \text n}[\ \text{log}_{\displaystyle \ \text n}(\sqrt[\displaystyle \text n]{{\sqrt[\displaystyle \text n]{\sqrt[\displaystyle \text n]{\sqrt[\displaystyle \text n]{\sqrt[\displaystyle \text n]{\text n }}}}}}})\ ] \\\\\\ -\text{log}_{\displaystyle \ \text n} [\ \text{log}_{\displaystyle \ \text n} \ \text n^{(\displaystyle \frac{1}{\text n^5})}\ ] \\\\\\ -\text{log}_{\displaystyle \ \text n}[\ \frac{1}{\text n^5}.\text{log}_{\displaystyle \text n}\ \text n \ ]$

$\displaystyle -\text{log}_{\displaystyle \ \text n}[\ \frac{1}{\text n^5}\ ] \\\\\\ -\text{log}_{\displaystyle \ \text n}\ \text n^{-5} = (-5).(-1).\text{log}_{\displaystyle\ \text n} \ \text n =(-5)(-1).1 = \ 5$

Portanto :

$\displaystyle -\text{log}_{\displaystyle \ \text n}[\ \text{log}_{\displaystyle \ \text n}(\sqrt[\displaystyle \text n]{{\sqrt[\displaystyle \text n]{\sqrt[\displaystyle \text n]{\sqrt[\displaystyle \text n]{\sqrt[\displaystyle \text n]{\text n }}}}}}})\ ] =\huge \boxed{ 5 }\checkmark$

Answer 2

Resposta:

\displaystyle -\text{log}_{\displaystyle \ \text n}[\ \text{log}_{\displaystyle \ \text n}(\sqrt[\displaystyle \text n]{{\sqrt[\displaystyle \text n]{\sqrt[\displaystyle \text n]{\sqrt[\displaystyle \text n]{\sqrt[\displaystyle \text n]{\text n }}}}}}})\ ] =\huge \boxed{ 5 }\checkmark

Explicação passo-a-passo:

ñ tenho certeza