Question

Log1/3 (81) - log1/3 (3) Como calcular

Answer 1

Lembrando que

$\large\fbox{ \ell og_a\begin{pmatrix}\dfrac{b}{c}\end{pmatrix}~\leftrightarrow~\ell og_a(b)-\ell og_a(c) }~~~~(0\ \textless \ a \neq 1)$

Desenvolvendo a expressão...

$\large\begin{array}{l}\ell og_{\frac{1}{3}}(81)-\ell og_{\frac{1}{3}}(3)~\leftrightarrow~\ell og_{\frac{1}{3}}\begin{pmatrix}\dfrac{81}{3}\end{pmatrix}\\\\\\\ell og_{\frac{1}{3}}(27)~\leftrightarrow~\ell og_{3^{-1}}(3^3)\end{array}$

Antes de prosseguir, atente-se a duas propriedades muito importantes:

$\large\fbox{ \ell og_a(b^c)~\leftrightarrow~c\cdot \ell og_a(b) }~~~~(0\ \textless \ a\neq1)\\\\\large\fbox{ \ell og_{a^{c}}(b)~\leftrightarrow~\frac{1}{c}\cdot\ell og_a(b) }~~~~(0\ \textless \ a\neq1)$

Continuando o desenvolvimento...

$\large\begin{array}{l}\ell og_{3^{-1}}(3^3)~\leftrightarrow~3\cdot\dfrac{~~1}{-1}\cdot\ell og_{3}(3)\\\\\\-3\ell og_3(3)~\leftrightarrow~-3\cdot1=-3\\\\\\\therefore~\fbox{ [\ell og_{\frac{1}{3}}(81)-\ell og_{\frac{1}{3}}(3)]=-3 }~~\leftarrow~resposta~(-3)\end{array}$

Obs: se o logaritmando é igual a base, o resultado do logaritmo é 1.