Question

Qual a resposta??
$\left[4ln4-4\right]-\left[1ln1-1\right]$

Se que a resposta é (8ln2-3), mas como chegar até lá?

Answer 1

Vamos usar a propriedades de Log :

$\text{ln} \ (\text a) = \text{Log}_{\ \displaystyle \text e} \ (\text a)$

$\text{ln(a)}^{\text k } = \text{ k . ln (a)}$

Temos :

$[\ 4.\text{ln(4)-4} \ ]- [ \ \text{ln(1)-1 \ ] }$

$[\ 4.\text{ln(2)}^2 -4} \ ]- [ \ \text{ln(1)-1 \ ] }$

sabemos que :

$\text{ln(1)} = \text k\to \text e^{\text k} = 1 \to \text e^{\text k}= \text e^{0} \to \boxed{\text k = 0}$

Então :

$[\ 4.\text{ln(2)}^2 -4} \ ]- [ \ \text{ln(1)-1 \ ] } \\\\ 4.2.\text{ln(2)}-4 - [ \ 0 - 1 \ ] \\\\ 8.\text{ln(2)} - 4 -(-1) \\\\ 8\text{ln(2)} -4+1 \\\\ \huge\boxed{8.\text{ln(2)} -3}\checkmark$