Question

3)determine o valor de X sabendo que
$log \frac{1}{8} \sqrt{2}^{3} = x$
pfvvvvvv alguém me ajuda​

Answer 1

• O resultado desse logaritmo é -1/2.

Bom, para resolver sua questão devemos aplicar a definição de log. Dada por: $\large\displaystyle\begin{aligned} \log _a b = x \end{aligned}$ , sendo a^{x} = b. Devemos também sempre ficar ligado nas condições de existência.

$\large\displaystyle\begin{aligned} \log_a b = x \Rightarrow\begin{cases} b\ n\tilde{a}o \ pode\ ser \ b\leq 0\\ a\ tem\ que\ ser > 0\ e\neq1\\ x = logaritmo\end{cases} \end{aligned}$

Sabendo disso, iremos aplicar a definição de log $\large\displaystyle\begin{aligned} \log _a b = x \end{aligned}$ e resolver seu probleminha.

$\large\displaystyle\text{ \begin{aligned} \log _{\frac{1}{ 8}} \sqrt{2}^3 = x\end{aligned} }$

$\large\displaystyle\text{ \begin{aligned} \log _{\frac{1}{ 8}} 2^{\frac{3}{2}} = x\end{aligned} }$

$\large\displaystyle\text{ \begin{aligned} 2^{\frac{3}{2}} = \left[\frac{1}{8} \right]^x\end{aligned} }$

$\large\displaystyle\text{ \begin{aligned} 2^{\frac{3}{2}} = \left[8^{-1}\right]^x\end{aligned} }$

$\large\displaystyle\text{ \begin{aligned} 2^{\frac{3}{2}} = \left[8\right]^{-x}\end{aligned} }$

$\large\displaystyle\text{ \begin{aligned} 2^{\frac{3}{2}} = \left[2^3\right]^{-x}\end{aligned} }$

$\large\displaystyle\text{ \begin{aligned} 2^{\frac{3}{2}} = 2^{-3x}\end{aligned} }$

Tendo então as bases iguais, basta cortamos as bases e trabalharmos apenas com os expoentes.

$\large\displaystyle\text{ \begin{aligned} \not{2}^{\frac{3}{2}} = \not{2}^{-3x}\end{aligned} }$

$\large\displaystyle\begin{aligned} \frac{3}{2}= -3x\end{aligned}$

$\large\displaystyle\begin{aligned} -3x=\frac{3}{2}\end{aligned}$

$\large\displaystyle\begin{aligned} \left[ -3x=\frac{3}{2}\right]\ \ \cdot \ \ (-1)\end{aligned}$

$\large\displaystyle\begin{aligned} 3x=-\frac{3}{2}\end{aligned}$

$\large\displaystyle\begin{aligned} x=-\frac{1}{2}\end{aligned}$

$\large\displaystyle\text{ \begin{aligned} \therefore \boxed{\boxed{\green{ \log_{\frac{1}{8}} \sqrt{2}^3=-\frac{1}{2}}}}\end{aligned} }$

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