Question

Se log de a na base ab=4, calcule:
$log_{ab} \frac{ \sqrt[3]{a} }{ \sqrt{b} }$
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Answer 1

Primeiro desenvolvemos o logaritmo descrito no começo:

$\log_{ab}a=4$

$(ab)^4=a$

$ab=\sqrt[4]{a}$

$b=\frac{\sqrt[4]{a} }{a}$

$b=a^{1/4-1}$

$b=a^{1/4-4/4}$

$b=a^{-3/4}$

Agora podemos calcular o logaritmo pedido:

$\log_{ab}\frac{\sqrt[3]{a} }{\sqrt{b} }=$

$\log_{ab}\sqrt[3]{a}-\log_{ab}\sqrt{b}=$

$\log_{ab}a^{1/3}-\log_{ab}b^{1/2}=$

$\log_{ab}a^{1/3}-\log_{ab}(a^{-3/4})^{1/2}=$

$\log_{ab}a^{1/3}-\log_{ab}a^{-3/8}=$

$\frac{\log_{ab}a}{3}-\frac{-3\log_{ab}a}{8}=$

$\frac{4}{3}-\frac{-3.4}{8}=$

$\frac{4}{3}-\frac{-12}{8}=$

$\frac{4}{3}+\frac{3}{2}=$

$\frac{8}{6}+\frac{9}{6}=$

$\frac{17}{6}$