Question

Resolva: Calcule o valor da expressão E ​

Answer 1

Resposta:

Explicação passo-a-passo:

Vamos fazer cada parcela separadamente

1ª parcela

$log_{\dfrac{1}{3}}81=log_{3^{-1}}3^{4}\\\\Propriedade\\\\log_{x^{y}}a^{b}=\dfrac{b}{y}.log_xa\\\\Aplicando\\\\log_{3^{-1}}3^{4}=\dfrac{4}{-1}log_33=-4.1=-4$

2ª parcela

$0,125=\dfrac{125}{1000}=\dfrac{1}{8}=\dfrac{1}{2^{3}}=2^{-3}\\ \\32=2^{5}\\\\\\log_{32}0,125=log_{2^{5}}2^{-3}=\dfrac{-3}{5}.log_22=\dfrac{-3}{5}.1=\dfrac{-3}{5}$

3ª parcela

$log\sqrt[3]{10}=log10^{\frac{1}{3}}=\dfrac{1}{3}.log10=\dfrac{1}{3}.1=\dfrac{1}{3}$

Somando as 3 parcelas

$E=-4+(\dfrac{-3}{5})+\dfrac{1}{3}\\\\\\mmc(3,5)=15\\\\E=\dfrac{-4.15-3.3+1.5}{15}\\\\\\E=\dfrac{-60-9+5}{15}\\\\\\E=\dfrac{-69+5}{15}\\\\\\\boxed{E=\dfrac{-64}{15}}$