Question

Qual e o valor de:
a)log2/3 8/27
b)log2 √2
c) log1/2 64
d) log2 √8

Answer 1

$a)\\\\log_{_\frac{2}{3}}\frac{8}{27}~=~x\\\\\\\frac{8}{27}~=~\left(\frac{2}{3}\right)^x\\\\\\\frac{2^3}{3^3}~=~\left(\frac{2}{3}\right)^x\\\\\\\left(\frac{2}{3}\right)^3~=~\left(\frac{2}{3}\right)^x\\\\\\\boxed{x~=~3}$

$b)\\\\log_{_2}\sqrt{2}~=~y\\\\\\\sqrt{2}~=~2^y\\\\\\2^{\frac{1}{2}}~=~2^y\\\\\\\boxed{y~=~\frac{1}{2}}$

$c)\\\\log_{_\frac{1}{2}}64~=~z\\\\\\64~=~\left(\frac{1}{2}\right)^z\\\\\\2^6~=~\left(2^{-1}\right)^z\\\\\\2^6~=~2^{\,-1~.~z}\\\\\\2^6~=~2^{-z}\\\\\\-z~=~6\\\\\\\boxed{z~=~-6}$

$d)\\\\log_{_2}\sqrt{8}~=~w\\\\\\\sqrt{8}~=~2^w\\\\\\\sqrt{2^3}~=~2^w\\\\\\2^{\frac{3}{2}}~=~2^w\\\\\\\boxed{w~=~\frac{3}{2}}$