Question

qual o resultado de:
log2 raiz de 8

Answer 1

Qual o resultado de:log2 raiz de 8

log 2 = x  ==> (√8)^x = 2 ==> (8^1/2)^x = 2^1
     √8

8^x/2 = 2^1 ==> (2^3)^x^/2 = 2^1

2^3x/2 = 2^1 ==> 3x/2 = 1

3x = 2 ==> x = 2/3

Answer 2

$log_{2}(\sqrt{8})=log_{2}(\sqrt[2]{2^{3}})$

Usando $\sqrt[b]{x^{a}}=x^{a/b}$, temos

$log_{2}(\sqrt{8})=log_{2}(2^{3/2})$

Usando a propriedade $log_{b}(a^{n})=n\cdot log_{b}(a)$:

$log_{2}(\sqrt{8})=\dfrac{3}{2}log_{2}(2)$

Como $log_{a}(a)=1~~para~todo~a~\textgreater~0,~a\neq1$:

$log_{2}(\sqrt{8})=\dfrac{3}{2}\cdot1\\\\\\\boxed{\boxed{log_{2}(\sqrt{8})=\dfrac{3}{2}}}$