Question

Resolva :
Log 3√3 de base 1/9

Answer 1

Puts . Tava estudando um dia desses,mais ja esqueci .. Affts trágico

Answer 2

Pela definição de logaritmo,

$\log_{\frac{1}{9}}3\sqrt{3}=x~~~\Leftrightarrow~~\big(\frac{1}{9}\big)^{x}=3\sqrt{3}$

Vamos resolver essa equação exponencial e encontrar $x$:

$\bigg(\dfrac{1}{9}\bigg)^{x}=3\sqrt{3}\\\\\\\bigg(\dfrac{1}{3^{2}}\bigg)^{x}=3^{1}\cdot\sqrt[2]{3^{1}}\\\\\\\big(3^{-2}\big)^{x}=3^{1}\cdot3^{1/2}\\\\\\3^{(-2)\cdot x}=3^{1+(1/2)}\\\\\\3^{-2x}=3^{3/2}$

As bases são números positivos e diferentes de 1, então podemos igualar os expoentes:

$-2x=\dfrac{3}{2}\\\\\\2x=-\dfrac{3}{2}\\\\\\x=-\dfrac{3}{2}\cdot\dfrac{1}{2}\\\\\\\boxed{\boxed{x=-\dfrac{3}{4}}}$
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Logo:

$\log_{\frac{1}{9}}3\sqrt{3}=x~~~~~\Longrightarrow~~~~~\boxed{\boxed{\log_{\frac{1}{9}}3\sqrt{3}=-\dfrac{3}{4}}}$