簡約-(5/6)log_{2}(5/6)-(1/6)log_{2}(1/6)

解

簡約

- (\frac{5}{6}) \cdot lo g_{2} (\frac{5}{6}) - (\frac{1}{6}) \cdot lo g_{2} (\frac{1}{6})

- \frac{1}{6} lo g_{2} (\frac{3125}{46656})

十進法表記

0.65002 \dots

解答ステップ

- (\frac{5}{6}) lo g_{2} (\frac{5}{6}) - (\frac{1}{6}) lo g_{2} (\frac{1}{6})

対数の規則を適用する:

a lo g_{c} (b) = lo g_{c} (b^{a})

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

= - lo g_{2} ((\frac{5}{6})^{(\frac{5}{6})}) - (\frac{1}{6}) lo g_{2} (\frac{1}{6})

対数の規則を適用する:

a lo g_{c} (b) = lo g_{c} (b^{a})

(\frac{1}{6}) lo g_{2} (\frac{1}{6}) = lo g_{2} ((\frac{1}{6})^{(\frac{1}{6})})

= - lo g_{2} ((\frac{5}{6})^{(\frac{5}{6})}) - lo g_{2} ((\frac{1}{6})^{(\frac{1}{6})})

- lo g_{2} ((\frac{5}{6})^{(\frac{5}{6})}) - lo g_{2} ((\frac{1}{6})^{(\frac{1}{6})})

を書き換え

- (lo g_{2} ((\frac{5}{6})^{(\frac{5}{6})}) + lo g_{2} ((\frac{1}{6})^{(\frac{1}{6})}))

= - (lo g_{2} ((\frac{5}{6})^{(\frac{5}{6})}) + lo g_{2} ((\frac{1}{6})^{(\frac{1}{6})}))

対数の規則を適用する:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

lo g_{2} ((\frac{5}{6})^{(\frac{5}{6})}) + lo g_{2} ((\frac{1}{6})^{(\frac{1}{6})}) = lo g_{2} ((\frac{5}{6})^{\frac{5}{6}} (\frac{1}{6})^{\frac{1}{6}})

= - lo g_{2} ((\frac{5}{6})^{\frac{5}{6}} (\frac{1}{6})^{\frac{1}{6}})

簡素化

lo g_{2} ((\frac{5}{6})^{\frac{5}{6}} (\frac{1}{6})^{\frac{1}{6}}) : \frac{1}{6} lo g_{2} ((\frac{5}{6})^{5} \frac{1}{6})

= - \frac{1}{6} lo g_{2} ((\frac{5}{6})^{5} \frac{1}{6})

(\frac{5}{6})^{5} = \frac{5 ^{5}}{6 ^{5}}

= - \frac{1}{6} lo g_{2} (\frac{1}{6} \cdot \frac{5 ^{5}}{6 ^{5}})

乗じる

\frac{5 ^{5}}{6 ^{5}} \cdot \frac{1}{6} : \frac{3125}{46656}

= - \frac{1}{6} lo g_{2} (\frac{3125}{46656})