Question

5. Se log x = 3 + log 3 - log 2 - 2 log 5, então
x é igual a:
A) 18
B) 25
C) 30
D) 60
E) 40​

Answer 1

Resposta:

$\boxed{\mathtt{D}}$

Explicação passo-a-passo:

$\\ \displaystyle \mathsf{\log x = 3 + \log 3 - \log 2 - 2 \cdot \log 5} \\\\ \mathsf{\log_{10} x = 3 \cdot \log_{10} 10 + \log_{10} 3 - \log_{10} 2 - 2 \cdot \log_{10} 5} \\\\ \mathsf{\log x = \left ( \log 10^3 + \log 3 \right ) - \left ( \log 2 + \log 5^2 \right )} \\\\ \mathsf{\log x = \log \left ( 10^3 \cdot 3 \right ) - \log \left ( 2 \cdot 5^2 \right )} \\\\ \mathsf{\log x = \log 3000 - \log 50} \\\\ \mathsf{\log x = \log \left ( \frac{3000}{50} \right )} \\\\ \mathsf{\log x = \log 60}$

Logo,

$\displaystyle \boxed{\boxed{\mathsf{x = 60}}}$

Propriedades aplicadas:

$\\ \displaystyle \mathtt{\bullet \qquad \log \left ( a \cdot b \right ) = \log a + \log b} \\\\ \mathtt{\bullet \qquad \log \left ( \frac{a}{b} \right ) = \log a - \log b} \\\\ \mathtt{\bullet \qquad \log \left ( a \cdot b \right ) = \log a + \log b} \\\\ \mathtt{\bullet \qquad c \cdot \log a = \log a^c}$

Onde, $\displaystyle \mathtt{a, b \in \mathbb{R_{+}^{\ast}}}$