Question

Log de 27 na base 3 raíz de 3. Gabarito=2

Answer 1

$\large\textsf{Principais propriedades dos logaritmos:}\\\\\\\mathsf{P1.~~\ell og_a(b\cdot c)=\ell og_a b+\ell og_a c}\\\\\\\mathsf{P2.~~\ell og_a(\dfrac{b}{c})=\ell og_ab-\ell og_ac}\\\\\\\mathsf{P3.~~\ell og_ab^{\alpha}=\alpha\cdot\ell og_ab}\\\\\\\textsf{Mudan\c{c}a~de~base:}~~\fbox{ \mathsf{\ell og_ab=\dfrac{\ell og_ca}{\ell og_cb}} }$


$\large\textsf{Principais propriedades exponenciais:}\\\\\\\mathsf{P1.~~a^m\cdot a^n=a^{m+n}}\\\\\\\mathsf{P2.~~\dfrac{a^m}{a^n}=a^{m-n}~~~~(para~a \neq0~e~m \geq n)}\\\\\\\mathsf{P3.~~(a\cdot b)^n=a^n\cdot b^n}\\\\\\\mathsf{P4.~~(\dfrac{a}{b})^n=\dfrac{a^n}{b^n}~~~~(b \neq 0)}\\\\\\\mathsf{P5.~~(a^m)^n=a^{m\cdot n}}\\\\\\\textsf{Expoente racional:}~~\large\fbox{ \mathsf{a^{\frac{b}{c}}=\sqrt[c]{\mathsf{a^b}}} }\\\\\\\textsf{Expoente inteiro negativo:}~~\fbox{ \mathsf{a^{-n}=\dfrac{1}{a^n}} }$


$\large\textsf{~Ap\'os esse resuminho podemos atacar a quest\~ao:}\\\\\\\\\begin{array}{l}\mathsf{\ell og_{3 \sqrt{3} }~(27)}~\Rightarrow~\mathsf{\ell og_{ \sqrt{3^2\cdot3}}~(27)~\Rightarrow~\ell og_{\sqrt{27}}~(27)}\\\\\\\mathsf{\ell og_{\sqrt{27}}~(27)=\dfrac{\ell og_3~27}{\ell og_3~\sqrt{27}}}~~~~\textsf{(mudan\c{c}a de base)}\end{array}$


$\large\begin{array}{l}\mathsf{ \dfrac{\ell og_3~27}{\ell og_3~27^\frac{1}{2}}~\Rightarrow~\dfrac{\ell og_3~27}{\frac{1}{2}\ell og_3~27}~\Rightarrow~\dfrac{\ell og_3~3^3}{\frac{1}{2}\ell og_3~3^3}~\Rightarrow~\dfrac{3\ell og_3~3}{\frac{3}{2}\ell og_3~3}}\\\\\\\mathsf{\dfrac{3\cdot 1}{\dfrac{3}{2}\cdot 1}~\Rightarrow~\dfrac{3}{\dfrac{3}{2}}~\Rightarrow~3\cdot\dfrac{2}{3}~\Rightarrow~\dfrac{6}{3}=2}\\\\\\\therefore~~\large\fbox{ \mathsf{\ell og_{3\sqrt{3}}~27=2~ }}\end{array}$


$\large\textsf{Prontinho, qualquer d\'uvida pode me notificar. =)}$