簡約-5/14log_{2}(5/14)-9/14log_{2}(9/14)

解

簡約

- \frac{5}{14} \cdot lo g_{2} (\frac{5}{14}) - \frac{9}{14} \cdot lo g_{2} (\frac{9}{14})

- \frac{1}{14} lo g_{2} (\frac{1210689028125}{1 4 ^{14}})

十進法表記

0.94028 \dots

解答ステップ

- \frac{5}{14} lo g_{2} (\frac{5}{14}) - \frac{9}{14} lo g_{2} (\frac{9}{14})

対数の規則を適用する:

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

\frac{5}{14} lo g_{2} (\frac{5}{14}) = lo g_{2} ((\frac{5}{14})^{\frac{5}{14}})

= - lo g_{2} ((\frac{5}{14})^{\frac{5}{14}}) - \frac{9}{14} lo g_{2} (\frac{9}{14})

対数の規則を適用する:

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

\frac{9}{14} lo g_{2} (\frac{9}{14}) = lo g_{2} ((\frac{9}{14})^{\frac{9}{14}})

= - lo g_{2} ((\frac{5}{14})^{\frac{5}{14}}) - lo g_{2} ((\frac{9}{14})^{\frac{9}{14}})

- lo g_{2} ((\frac{5}{14})^{\frac{5}{14}}) - lo g_{2} ((\frac{9}{14})^{\frac{9}{14}})

を書き換え

- (lo g_{2} ((\frac{5}{14})^{\frac{5}{14}}) + lo g_{2} ((\frac{9}{14})^{\frac{9}{14}}))

= - (lo g_{2} ((\frac{5}{14})^{\frac{5}{14}}) + lo g_{2} ((\frac{9}{14})^{\frac{9}{14}}))

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}{14})^{\frac{5}{14}}) + lo g_{2} ((\frac{9}{14})^{\frac{9}{14}}) = lo g_{2} ((\frac{5}{14})^{\frac{5}{14}} (\frac{9}{14})^{\frac{9}{14}})

= - lo g_{2} ((\frac{5}{14})^{\frac{5}{14}} (\frac{9}{14})^{\frac{9}{14}})

簡素化

lo g_{2} ((\frac{5}{14})^{\frac{5}{14}} (\frac{9}{14})^{\frac{9}{14}}) : \frac{1}{14} lo g_{2} ((\frac{9}{14})^{9} (\frac{5}{14})^{5})

= - \frac{1}{14} lo g_{2} ((\frac{9}{14})^{9} (\frac{5}{14})^{5})

(\frac{5}{14})^{5} = \frac{5 ^{5}}{1 4 ^{5}}

= - \frac{1}{14} lo g_{2} ((\frac{9}{14})^{9} \frac{5 ^{5}}{1 4 ^{5}})

(\frac{9}{14})^{9} = \frac{9 ^{9}}{1 4 ^{9}}

= - \frac{1}{14} lo g_{2} (\frac{5 ^{5}}{1 4 ^{5}} \cdot \frac{9 ^{9}}{1 4 ^{9}})

乗じる

\frac{5 ^{5}}{1 4 ^{5}} \cdot \frac{9 ^{9}}{1 4 ^{9}} : \frac{1210689028125}{1 4 ^{14}}

= - \frac{1}{14} lo g_{2} (\frac{1210689028125}{1 4 ^{14}})