Question

3. Calcule o valor de S: S = log8 √2 + log√2 8 − log√2 √8
(considere a screenshot por favor)

Answer 1

Propriedades usadas

$\sqrt{a}=a^{\frac{1}{2}}\\\log_{a^{\frac{c}{d}}}b=\dfrac{d}{c}\cdot\log_{a}b\\\\\log_ab^c=c\cdot\log_ab\\\\\log_aa=1$

Solução:

$S=\log_8\sqrt{2}+\log_{\sqrt{2}}8-\log_{\sqrt{2}}\sqrt{8}\\\\S=\log_{2^3}2^{\frac{1}{2}}+\log_{2^{\frac{1}{2}}}2^3-\log_{2^{\frac{1}{2}}}(2^3)^{\frac{1}{2}}\\\\S=\dfrac{1}{3}\cdot\dfrac{1}{2}\log_22+2\cdot3\log_22-2\cdot3\cdot\dfrac{1}{2}\log_22\\\\S=\dfrac{1}{6}+6-\dfrac{6}{2}\\\\S=\dfrac{1}{6}+6-3\\\\S=\dfrac{1}{6}+3\\\\S=\dfrac{1+18}{6}\\\\\boxed{\boxed{S=\dfrac{19}{6}}}$