Question

Log de $3/3 \sqrt{3}$ na base $\sqrt[4]{3}$

Answer 1

OBS: O índice de 3√3 é 3 e não 2!

$\log_{ \sqrt[4]{3} } \frac{3}{3 \sqrt[3]{3} } = \log_{ \sqrt[4]{3} } 3 - \log_{ \sqrt[4]{3} } \sqrt[3]{3} }$

Resolvendo os log separadamente: 

$\log_{ \sqrt[4]{3} } 3=x \\ 3^{ \frac{x}{4} } =3\\ \frac{x}{4}=1\\ x=4$

$\log_{ \sqrt[4]{3} } 3 \sqrt[3]{3}} =y$
$3^{\frac{y}{4}} =3^{ \frac{1}{3} }$
$\frac{y}{4}= \frac{1}{3} \\ y= \frac{4}{3}$

$\log_{ \sqrt[4]{3} } \frac{3}{3 \sqrt[3]{3}} = 4- \frac{4}{3} = \frac{8}{3}$

Answer 2

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$\log_{ \sqrt[4]{3} }{ \frac{3}{ \sqrt[3]{3} } = \\ \\ \\ ( \sqrt[4]{3})^x= \frac{3}{ \sqrt[3]{3} }$

$(3 ^{ \frac{1}{4} )^x= \frac{3}{3^ \frac{1}{3} }$

$3^{ \frac{x}{4}}=3\div3^{ \frac{1}{3}$

$3^{ \frac{x}{4} }=3^{ \frac{3-1}{3}$

$3^{ \frac{x}{4} }=3^{ \frac{2}{3}$

$\frac{x}{4} = \frac{2}{3} \\ \\ 3x=8 \\ \\ x= \frac{8}{3}$