Question

O número real x, positivo ...

Answer 1

Resposta:

Propriedades usadas:

$log_{a}( \sqrt{b} ) = \frac{1}{2} log_{a}(b)$

$log_{a}( {b}^{ \alpha } ) = \alpha . log_{a}(b)$

$1 = log_{a}(a)$

$log_{a}( \frac{b}{c} ) = log_{a}(b) - log_{a}(c)$

${a}^{ log_{a}(b) } = b$

Com isso, temos:

$log_{x}(2x) log_{2}(x) = 3 - \frac{1}{2} log_{2}(x)$

Multiplicando os lados por 2, temos:

$2 log_{x}(2x) log_{2}(x) = 6 - log_{2}(x)$

$log_{2}( {x}^{2 log_{x}(2x) } ) = log_{2}( {2}^{6} ) - log_{2}(x)$

$log_{2}( {x}^{2 log_{x}(2x) } ) = log_{2}( \frac{ {2}^{6} }{x} )$

${x}^{2 log_{x}(2x) } = \frac{64}{x}$

${x}^{ log_{x}( {2x}^{2} ) } = \frac{64}{x}$

$( {2x})^{2} = \frac{64}{x}$

$4 {x}^{2} = \frac{64}{x}$

$4 {x}^{3} = 64$

${x}^{3} = 16$

$x = 2 \sqrt[3]{2}$

Letra C.