Question

Encontre x= log150 de base 200

Answer 1

$\mathsf{\log_{200}\,\,150=x}\\ \\ \\ \mathsf{\dfrac{\log\,\,150}{\big\log\,\,200}=x}\\ \\ \\ \mathsf{\dfrac{\log\,\,(3\,.\,5\,.\,10)}{\log\,\,(2\,.\,100)}=x}\\ \\ \\ \mathsf{\dfrac{\log\,\,3\,+\,\log\,\,5\,+\,\log\,\,10}{\log\,\,2\,+\,\log\,\,100}=x}$

Sabendo que 5 = 10/2, log 10 = 1 e log 100 = 2, temos:

$\mathsf{\dfrac{\log\,\,3\,+\,\log\,\,\bigg(\dfrac{10}{2}\bigg)\,+\,1}{\log\,\,2\,+\,2}=x}\\ \\ \\ \mathsf{\dfrac{\log\,\,3\,+\,\big[\log\,\,10\,-\,\log\,\,2\big]\,+\,1}{\log\,\,2\,+\,2}=x}\\ \\ \\ \mathsf{\dfrac{\log\,\,3\,+\m1\,-\,\log\,\,2\,+\,1}{\log\,\,2\,+\,2}=x}\\ \\ \\ \boxed{\boxed{\mathsf{\dfrac{\log\,\,3\,-\,\log\,\,2\,+\,2}{\log\,\,2\,+\,2}=x}}}$

Na maioria dos casos devemos simplificar até deixar tudo em log 2 e log 3.