Question

log1/4 na base 2 raiz de 2

Answer 1

$log_{(2\sqrt{2})}\left(\dfrac{1}{4}\right)=x$

Aplicando a definição de logaritmos:

$(2\sqrt{2})^{x}=\dfrac{1}{4}\\\\\\(2\cdot2^{1/2})^{x}=\dfrac{1}{2^{2}}$

Invertendo a base e trocando o sinal do expoente:

$(2^{1}\cdot2^{1/2})^{x}=2^{-2}\\(2^{[1+(1/2)]})^{x}=2^{-2}\\(2^{3/2})^{x}=2^{-2}\\2^{3x/2}=2^{-2}$

Bases iguais, iguale os expoentes:

$\dfrac{3x}{2}=-2\\\\\\3x=-2\cdot2\\\\\\3x=-4\\\\\\\boxed{\boxed{x=-\dfrac{4}{3}}}$