簡約-4/10 log_{2}(4/10)-6/10 log_{2}(6/10)

解

簡約

- \frac{4}{10} lo g_{2} (\frac{4}{10}) - \frac{6}{10} lo g_{2} (\frac{6}{10})

- lo g_{2} (5 \frac{27}{125} 5 \frac{4}{25})

十進法表記

0.97095 \dots

解答ステップ

- \frac{4}{10} lo g_{2} (\frac{4}{10}) - \frac{6}{10} lo g_{2} (\frac{6}{10})

- \frac{4}{10} lo g_{2} (\frac{4}{10}) = - \frac{2}{5} lo g_{2} (\frac{2}{5})

\frac{6}{10} lo g_{2} (\frac{6}{10}) = \frac{3}{5} lo g_{2} (\frac{3}{5})

= - \frac{2}{5} lo g_{2} (\frac{2}{5}) - \frac{3}{5} lo g_{2} (\frac{3}{5})

- \frac{2}{5} lo g_{2} (\frac{2}{5}) = - lo g_{2} ((\frac{2}{5})^{\frac{2}{5}})

- \frac{3}{5} lo g_{2} (\frac{3}{5}) = - lo g_{2} ((\frac{3}{5})^{\frac{3}{5}})

= - lo g_{2} ((\frac{2}{5})^{\frac{2}{5}}) - lo g_{2} ((\frac{3}{5})^{\frac{3}{5}})

対数の規則を適用する:

lo g_{a} (x) + lo g_{a} (y) = lo g_{a} (x y)

- lo g_{2} ((\frac{3}{5})^{\frac{3}{5}}) - lo g_{2} ((\frac{2}{5})^{\frac{2}{5}}) = lo g_{2} ((\frac{3}{5})^{\frac{3}{5}} (\frac{2}{5})^{\frac{2}{5}})

= - lo g_{2} ((\frac{3}{5})^{\frac{3}{5}} (\frac{2}{5})^{\frac{2}{5}})

(\frac{3}{5})^{\frac{3}{5}} (\frac{2}{5})^{\frac{2}{5}} = 5 \frac{27}{125} 5 \frac{4}{25}

= - lo g_{2} (5 \frac{27}{125} 5 \frac{4}{25})