Question

Encontre o valor de log(72⋅√15). Usando propriedades de logaritmo. Detalhe sua resolução.
Considere: log2=0,3 , log3=0,48 e log5=0,7

Me ajudemmmmmmm pfvr estou desesperada

Answer 1

Vamos usar as propriedades de Log :

$\text{Log(a.b)} = \text{Log(a)}+\text{Log(b)}$

$\text{Log(a)}^{\text n} = \text n.\text{Log(a)}$

Temos :

$\text{Log}{(72\sqrt{15})}$

aplicando as propriedades de Log :

$\displaystyle \text{Log(72)+Log}(\sqrt{15}) \\\\ \text{Log(8.9)}+\text{Log(15)}^{\frac{1}{2}} \\\\ \text{Log(8)+\text{Log(9)}} + \frac{1}{2}.\text{Log(15)} \\\\ \text{Log(2)}^3+\text{Log(3)}^2+\frac{\text{Log(5.3)}}{2} \\\\ 3.\text{Log(2)}+2.\text{Log(3)}+\frac{\text{Log(5)+\text{Log(3)}}}{2}$

A questão nos diz que  :

$\text{Log(2)}=0,3 \\\\ \text{Log(3)}=0,48 \\\\ \text{Log(5)}=0,7$

Substituindo os respectivos valores :

$\displaystyle 3.\text{Log(2)}+2.\text{Log(3)}+\frac{\text{Log(5)+\text{Log(3)}}}{2}$

$\displaystyle 3.(0,3)+2.(0,48)+\frac{\text{(0,7)+(0,48)}}{2}$

$\displaystyle 0,9+0,96+\frac{1,18}{2}$

$\displaystyle 1,86+0,59$

$2,45$

Portanto :

$\huge\boxed{\text{Log}{(72\sqrt{15})} = 2,45}\checkmark$