Question

SendoLog 2 = 0,3; Log 3 =0,4 e Log 5 = 0,7, Calcule Log_{2}50 ​

Answer 1

Neste exercício, vamos utilizar a fatoração e as propriedade logarítmicas.

$\log_{_2}50~=\\\\\\\underline{Fatorando}~o~logaritmando~50\\\\\\=~\log_{_2}\left(2\cdot5\cdot5\right)\\\\\\=~\log_{_2}\left(2\cdot5^2\right)\\\\\\Aplicando~a~propriedade~do~\underline{logaritmo~do~produto}\\\\\\=~\log_{_2}2~+~\log_{_2}5^2\\\\\\Aplicando~a~propriedade~do~\underline{logaritmo~da~potencia}\\\\\\=~\log_{_2}2~+~2\cdot\log_{_2}5$$Logaritmo~com~base~e~logaritmando~identicos~vale~1\\\\\\=~1~+~2\cdot\log_{_2}5\\\\\\Temos~apenas~valores~para~logaritmos~na~base~10,~logo~vamos\\aplicar~a~propriedade~da~troca~de~base~em~\log_{_2}5\\\\\\=~1~+~2\cdot\left(\dfrac{\log5}{\log2}\right)\\\\\\Subsituindo~os~valores~conhecidos\\\\\\=~1~+~2\cdot\left(\dfrac{0,7}{0,3}\right)\\\\\\=~1~+~\dfrac{2\cdot7}{3}\\\\\\=~\dfrac{3~+~14}{3}\\\\\\=~\boxed{\dfrac{17}{3}~~ou~~5,666...}$