Question

se 2^a=3, pode-se afirmar que (256)^a/2 corresponde a:
a) 81
b)27
c)12
d)3

Answer 1

Resposta:

$\boxed{\mathtt{A}}$

Explicação passo-a-passo:

De acordo com o enuciado,

$\displaystyle \mathtt{2^a = 3}$

Aplicando a definição de Logaritmos, teremos:

$\displaystyle \mathtt{2^a = 3 \Leftrightarrow \boxed{\mathtt{\log_2 3 = a}}}$

Por conseguinte, tomemos $\displaystyle \mathtt{256^{a/2} = \lambda}$

Com efeito,

$\\ \displaystyle \mathsf{256^{a/2} = \lambda} \\\\ \mathsf{\log_{256} \lambda = \frac{a}{2}} \\\\ \mathsf{a = 2 \cdot \log_{2^8} \lambda} \\\\ \mathsf{a = 2 \cdot \frac{1}{8} \cdot \log_2 \lambda} \\\\ \boxed{\mathsf{a = \frac{1}{4} \cdot \log_2 \lambda}}$

Comparando os a's,

$\\ \displaystyle \mathsf{a = a} \\\\ \mathsf{\log_2 3 = \frac{1}{4} \cdot \log_2 \lambda} \\\\ \mathsf{4 \cdot \log_2 3 = \log_2 \lambda} \\\\ \mathsf{\log_2 3^4 = \log_2 \lambda} \\\\ \mathsf{\log_2 81 = \log_2 \lambda} \\\\ \boxed{\boxed{\mathsf{\lambda = \boxed{\mathsf{(256)^{a/2} = 81}}}}}$