Question

por favor me ajudem , não consigo resover isso , e tem pouco tempo
$log_{3}(27 + log_{ \frac{1}{5} }(125) + log_{4}( \sqrt{32} ) + log_{ \frac{2}{3} }( \frac{8}{27} )$

Answer 1

A forma mais simples é calcular os logs e depois calcularmos a expressão:

$log_{_3}27=x\\\\27=3^x\\\\3^3=3^x\\\\x=3$

$log_{_{\frac{1}{5}}}125=a\\\\125=\left(\frac{1}{5}\right)^a\\\\5^3=\left(\frac{1}{5}\right)^a\\\\\left(\frac{1}{5}\right)^{-3}=\left(\frac{1}{5}\right)^a\\\\a=-3$

$log_{_4}\sqrt{32}=b\\\\\sqrt{32}=4^b\\\\32^{\frac{1}{2}}=\left(2^2\right)^b\\\\\left(2^5\right)^{\frac{1}{2}}=\left(2\right)^{\;2\;.\;b}\\\\\left(2\right)^{\;5\;.\;\frac{1}{2}}=\left(2\right)^{\;2\;.\;b}\\\\\left(2\right)^{\frac{5}{2}}=\left(2\right)^{\;2b}\\\\2b=\frac{5}{2}\\\\b=\frac{5}{4}$

$log_{_{\frac{2}{3}}}\frac{8}{27}=c\\\\\frac{8}{27}=\left(\frac{2}{3}\right)^{c}\\\\\left(\frac{2^3}{3^3}\right)=\left(\frac{2}{3}\right)^{c}\\\\\left(\frac{2}{3}\right)^{3}=\left(\frac{2}{3}\right)^{c}\\\\c=3$

Juntando tudo na expressão:

$x+a+b+c\\\\3+(-3)+\frac{5}{4}+3\\\\\frac{1\;.\;5+4\;.\;3}{4}\\\\\frac{17}{4}$

Resp: 17/4