Question

Calcule o valor da expressão log 0,5 na base 2 + log raiz de 3 na base 3 - log 8 na base 4

Answer 1

A forma mais simples de resolver é calcular separadamente os logaritmos e então calcular a expressão.

$log_{_2}0,5=a\\\\log_{_2}\frac{1}{2}=a\\\\\frac{1}{2}=2^{a}\\\\2^{-1}=2^a\\\\a=-1$

$log_{_3}\sqrt{3}=b\\\\log_{_3}3^{\frac{1}{2}}=b\\\\3^{\frac{1}{2}}=3^b\\\\b=\frac{1}{2}$

$log_{_4}8=c\\\\8=4^c\\\\2^3=\left(2^2\right)^c\\\\2^3=2^{\,2\;.\;c}\\\\2^3=2^{2c}\\\\2c=3\\\\c=\frac{3}{2}$

Agora calculando a expressão:

$log_{_2}0,5+log_{_3}\sqrt{3}-log_{_4}8\\\\a+b-c\\\\-1+\frac{1}{2}-\frac{3}{2}\\\\-1-\frac{3-1}{2}\\\\-1-\frac{2}{2}\\\\-1-1\\\\-2$

Resp: -2