Question

o gabarito esta dizendo q e a letra D) mas eu queria saber como resolve essa questão​

Answer 1

Vamos usar as seguintes propriedades :

• Logaritmando e base iguais :

              $\displaystyle \text{Log}_{\ \displaystyle \text x}\ (\text x) = 1$  

• Definição de Log :

              $\displaystyle \text{Log}_{\ \displaystyle \text y}\ (\text x) = \text p \to \text x = \text y^{\text p}$

• Expoente no Logaritmando :

              $\displaystyle \text{Log}_{\ \displaystyle \text y}\ (\text x)^{\text n} = \text{n.Log}_{\ \text y}(\text x)$

• Expoente na base :

             $\displaystyle \text{Log}_{\ \displaystyle \text y^{\displaystyle (\text n)}}\ (\text x) = \frac{1}{\text n}.\text{Log}_{\text y}.(\text x)$

Temos a seguinte questão.

$\text{a,b e c } \in \mathbb{R} > 1 \ (\text{ reais maiores que 1})$

* $\displaystyle \text{Log}_{\ \displaystyle \text b}\ (\text a ) = 2 \to \text a = \text b^2$

* $\displaystyle \text{Log}_{\ \displaystyle \text b}\ (\text c) = 3 \to \text c = \text b^3$

O valor da expressão :

$\displaystyle \text{Log}_{\ \displaystyle \text c}\ (\text{a.c}) + 3.\text{Log}_{\ \displaystyle \text c}\ (\text b) - 2.\text{Log}_{\ \displaystyle \text c}\ (\frac{\text a}{\text b})$

Substituindo os respectivos valores em função de b :

$\displaystyle \text{Log}_{\ \displaystyle \text b^3}\ (\text b^2.\text b^3) + 3.\text{Log}_{\ \displaystyle \text b^3}\ (\text b) - 2.\text{Log}_{\ \displaystyle \text b^3}\ (\frac{\text b^2}{\text b})$

$\displaystyle \text{Log}_{\ \displaystyle \text b^3}\ (\text b^5) + 3.\text{Log}_{\ \displaystyle \text b^3}\ (\text b) - 2.\text{Log}_{\ \displaystyle \text b^3}\ (\text b)$

Aplicando as propriedades :

$\displaystyle 5.\frac{1}{3}.\text{Log}_{\ \displaystyle \text b}\ (\text b) + 3.\frac{1}{3}.\text{Log}_{\ \displaystyle \text b}\ (\text b) - 2.\frac{1}{3}.\text{Log}_{\ \displaystyle \text b}\ (\text b)$

$\displaystyle \frac{5}{3} + \frac{3}{3}. - \frac{2}{3}$

$\displaystyle \frac{5-2}{3} + 1 \to \frac{3}{3} + 1 \to 1+1$

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