Question

Considere log 2 = a, 3 = b calcule:

a)log 25
B) log(8 . \Г27)​

Answer 1

a)

$log\,25~=\\\\\\=~log\,(5^2)~=\\\\\\=~log\,\left(\frac{10}{2}\right)^2~=\\\\\\Utilizando~a~propriedade~do~logaritmo~do~expoente:\\\\\\=~2.log\,\left(\frac{10}{2}\right)~=\\\\\\Utilizando~a~propriedade~do~logaritmo~do~quociente:\\\\\\2~.~(log\,10~-~log\,2)~=\\\\\\=~2~.~(1~-~a)~=\\\\\\=~\boxed{2-2a}$

b)

$log\,(8~.~\sqrt{27})~=\\\\\\Utilizando~a~propriedade~do~logaritmo~do~produto:\\\\\\=~log\,8~+~log\,\sqrt{27}~=\\\\\\=~log\,2^3~+~log\,\sqrt{3^3}~=\\\\\\=~log\,2^3~+~log\,3^{\frac{3}{2}}~=\\\\\\Utilizando~a~propriedade~do~logaritmo~do~expoente:\\\\\\=~3.log\,2~+~\frac{3}{2}.log\,3~=\\\\\\=~3~.~a~+~\frac{3}{2}~.~b~=\\\\\\=~\boxed{3a+\frac{3}{2}b}$