簡約-1/5 log_{2}(1/5)-4/5 log_{2}(4/5)

解

簡約

- \frac{1}{5} lo g_{2} (\frac{1}{5}) - \frac{4}{5} lo g_{2} (\frac{4}{5})

lo g_{2} (\frac{5 5 5 625}{2 5 8})

十進法表記

0.72192 \dots

解答ステップ

- \frac{1}{5} lo g_{2} (\frac{1}{5}) - \frac{4}{5} lo g_{2} (\frac{4}{5})

- \frac{1}{5} lo g_{2} (\frac{1}{5}) = \frac{1}{5} lo g_{2} (5)

= \frac{1}{5} lo g_{2} (5) - \frac{4}{5} lo g_{2} (\frac{4}{5})

\frac{1}{5} lo g_{2} (5) = lo g_{2} (5^{\frac{1}{5}})

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

= lo g_{2} (5^{\frac{1}{5}}) - lo g_{2} ((\frac{4}{5})^{\frac{4}{5}})

対数の規則を適用する:

lo g_{a} (x) - lo g_{a} (y) = lo g_{a} (\frac{x}{y})

lo g_{2} (5^{\frac{1}{5}}) - lo g_{2} ((\frac{4}{5})^{\frac{4}{5}}) = lo g_{2} \frac{5 ^{\frac{1}{5}}}{( \frac{4}{5} ) ^{\frac{4}{5}}}

= lo g_{2} \frac{5 ^{\frac{1}{5}}}{( \frac{4}{5} ) ^{\frac{4}{5}}}

\frac{5 ^{\frac{1}{5}}}{( \frac{4}{5} ) ^{\frac{4}{5}}} = \frac{5 ^{\frac{1}{5}} 5 625}{2 5 8}

= lo g_{2} (\frac{5 ^{\frac{1}{5}} 5 625}{2 5 8})

指数の規則を適用する:

a^{\frac{1}{n}} = n a

5^{\frac{1}{5}} = 55

= lo g_{2} (\frac{5 5 5 625}{2 5 8})