Question

(LOGARITMO) Calcule o valor de K:

K= 3.Log 2 na base 4 - 2/3. Log 10 na base 100 - 1/2 . Log 1/2 na base 4

Desde já agradeço.

Answer 1

$k = 3.log_{4}2- \frac{2}{3} log_{100}10- \frac{1}{2} log_{4} \frac{1}{2} \\ \\ 3.log_{4}2 = x \\ log_{4}2^{3}=x \\ 4^{x}=2^{3} \\ (2^{2})^{x}=2^{3} \\ 2^{2x}=2^3 \\ 2x = 3 \\ x= \frac{3}{2} \\ \\ \frac{2}{3} log_{100}10=y \\ log_{100}10^{ \frac{2}{3} }=y \\ 100^{y} = 10^{ \frac{2}{3} } \\ (10^{2})^{y}=10^{ \frac{2}{3} } \\ 10^{2y}=10^{ \frac{2}{3} } \\ 2y= \frac{2}{3} \\ y= \frac{2}{6} \\ \\ \frac{1}{2} log_{4} \frac{1}{2} = z \\log_{4}( \frac{1}{2} )^{\frac{1}{2}}=z$

$\\log_{4}( \frac{1}{2} )^{\frac{1}{2}}=z \\ 4^{z}=( \frac{1}{2} )^{\frac{1}{2}} \\ 2^{2z}= (\frac{1}{2})^{-2z} = ( \frac{1}{2} )^{ \frac{1}{2} } \\ -2z= \frac{1}{2} \\ z=- \frac{1}{4} \\ \\ k = x - y - z \\ \\ k = \frac{3}{2} - \frac{2}{6} - (- \frac{1}{4} ) \\ \\ k=\frac{3}{2} - \frac{2}{6}+ \frac{1}{4} \\ \\ k= \frac{36-8+6}{24} \\ \\ k = \frac{34}{24} = \frac{17}{12}$

Espero ter ajudado.