Question

Desenvolver aplicando as propriedades dos logaritmos (a, b e c são reais positivos):

㏒₂$\sqrt{ \frac{4a \sqrt{ab} }{b \sqrt[3]{a^{2}b } } }$

No gabarito a resposta está como:
$2+ \frac{5}{12}$㏒₂$a- \frac{5}{12}$㏒₂$b$

Answer 1

$\log_2\sqrt{ \frac{ 4a\sqrt{ab} }{ b\sqrt[3]{a^2 b} } } \\ \\ \log_2({ \frac{ \sqrt{16a^3 b} }{ \sqrt[3]{a^2 b^4} } )^\frac{1}{2}} \\ \\ \frac{1}{2} \log_2{ \frac{ \sqrt{16a^3 b} }{ \sqrt[3]{a^2 b^4} }} \\ \\ {\frac{1}{2} \log_2\sqrt{16a^3 b}}-\frac{1}{2} \log_2\sqrt[3]{a^2 b^4}} \\ \\ {\frac{1}{2} \log_2(16a^3 b)^ \frac{1}{2}}-{\frac{1}{2} \log_2(a^2 b^4})^ \frac{1}{3}} \\ \\ {\frac{1}{4} \log_2(16a^3 b)}-{\frac{1}{6} \log_2(a^2 b^4)}} \\ \\{ {\frac{1}{4} \log_2 2^4}+ {\frac{1}{4} \log_2 a^3}}+ \frac{1}{4}\log_2b}-( \frac{1}{6}\log_2a^2}+ \frac{1}{6}\log_2b^4)}}   \\  \\  {\frac{4}{4} \log_2 2}+ {\frac{3}{4} \log_2 a}}+ \frac{1}{4}\log_2b}- \frac{2}{6}\log_2a}- \frac{4}{6}\log_2b}  \\  \\ 1+( \frac{3}{4} - \frac{1}{3})\log_2a}+( \frac{1}{4}- \frac{2}{3})\log_2b} \\  \\ 1+ {\frac{5}{12}\log_2a}-{ \frac{5}{12}\log_2b$