Question

$log _{4} \sqrt[13]{64}$ = ???

Answer 1

Usando das seguintes propriedades:

$Expoente~fracion\'ario:~~~\large\fbox{ a^{\frac{x}{n}}~\Leftrightarrow~ \sqrt[n]{a^x} }\\\\\ Logaritmo~da~pot\^encia:~~\large\fbox{ \ell og_a(b^c)~\Leftrightarrow~c\cdot\ell og_a(b) }\\\\Decorrente~de~mudan\c{c}a~de~base~temos:~\large\fbox{ log_{a^b}(c)~\Leftrightarrow~\frac{1}{b}\cdot\ell og_a(c) }$

Portanto:

$\ell og_4(\sqrt[13]{64})~\Leftrightarrow~\ell og_4(64^{\frac{1}{13}})~\Leftrightarrow~\frac{1}{13}\cdot\ell og_4(64)\\\\\ \frac{1}{13}\ell og_{2^2}(64)~\Leftrightarrow~\frac{1}{2}\cdot\frac{1}{13}\cdot\ell og_2(64)~\Leftrightarrow~\frac{1}{26}\cdot\ell og_2(64)\\\\\frac{1}{26}\cdot\ell og_2(2^6)~\Leftrightarrow~6\cdot\frac{1}{26}\cdot\ell og_2(2)~\Leftrightarrow~\frac{6}{26}\cdot\ell og_{2{\hspace{-6}}\diagup}(2\hspace{-6}\diagup)$

$\frac{6}{26}\cdot1~\Leftrightarrow~\dfrac{6}{26}~\Leftrightarrow~\dfrac{6(\div 2)}{26(\div2)}~\Leftrightarrow~\fbox{ \dfrac{3}{13} }~~\checkmark$