Question

calcule a soma S em cada caso:
a) S= log 8 base 2 + log 1/9 na base 3 + log raiz quadrada de cinco na base 5


b) S= log 0,01 base 100 + log raiz cúbica de 5 na base 25 - log 2 na base raiz quadrada de 2

Answer 1

$a)log_{2}8+ log_{3} \frac{1}{9} + log_{5} \sqrt{5}$
$Resolvendo log_{2}8$
$2^{x}= 2^{3}$
$x=3$
Resolvendo $log_{3} \frac{1}{9}$
$log_{3} \frac{1}{ 3^{2} }$
$3^{x} = 3^{-2}$
$x=-2$
Resolvendo $log_{5} \sqrt{5}$
$5^{x} = 5^{ \frac{1}{2} }$
$x= \frac{1}{2}$

Portanto somando os resultados achados
$3+(-2)+ \frac{1}{2}$
S= $\frac{3}{2}$


b) $log_{100}0.01+log_{25} \sqrt[3]{5} -log_{2}2$
Resolvendo $log_{100}0.01$
$100^{x} = 100^{-1}$
$x=-1$
Resolvendo $log_{25} \sqrt[3]{5}$
$5^{2x}=5^{ \frac{1}{3} }$
$2x= \frac{1}{3}$
$x= \frac{1}{6}$
Resolvendo $log_{2}2$
$2^{ \frac{1}{2}x} =2^{1}$
$\frac{1}{2}x=1$
$x=2$

Entao resolvendo a equcao
$-4+ \frac{1}{6} - (2)$
S=$\frac{-17}{6}$