Question

Se log 2=x e log3=y , então 375

Answer 1

Dados

$\mathrm{\ell og\,}2=x\;\text{ e }\;\mathrm{\ell og\,}3=y,$

calcular $\mathrm{\ell og\,}375:$
 

Decompondo o número $375$ em seus fatores primos:

$\mathrm{\ell og\,}375=\mathrm{\ell og\,}(3\cdot 5^{3})$


Aplicando as propriedades operatórias dos logaritmos:

$\mathrm{\ell og\,}375=\mathrm{\ell og\,}3+\mathrm{\ell og\,}(5^{3})\\ \\ \mathrm{\ell og\,}375=\mathrm{\ell og\,}3+3\,\mathrm{\ell og\,}5$


Não temos o valor de $\mathrm{\ell og}\,5,$ mas podemos reescrever o $5$como $\dfrac{10}{2}:$

$\mathrm{\ell og\,}375=\mathrm{\ell og\,}3+3\,\mathrm{\ell og}\left(\dfrac{10}{2} \right )\\ \\ \\ \mathrm{\ell og\,}375=\mathrm{\ell og\,}3+3\,(\mathrm{\ell og\,}10-\mathrm{\ell og}\,2)\\ \\ \mathrm{\ell og\,}375=\mathrm{\ell og\,}3+3\,(1-\mathrm{\ell og}\,2)$


Substituindo os valores dados,

$\mathrm{\ell og\,}375=y+3\,(1-x)\\ \\ \mathrm{\ell og\,}375=y+3-3x$