簡約-9/20log_{2}(9/20)-1/20log_{2}(1/20)

解

簡約

- \frac{9}{20} \cdot lo g_{2} (\frac{9}{20}) - \frac{1}{20} \cdot lo g_{2} (\frac{1}{20})

- \frac{1}{20} lo g_{2} (\frac{387420489}{2 0 ^{10}})

十進法表記

0.73449 \dots

解答ステップ

- \frac{9}{20} lo g_{2} (\frac{9}{20}) - \frac{1}{20} lo g_{2} (\frac{1}{20})

対数の規則を適用する:

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

\frac{9}{20} lo g_{2} (\frac{9}{20}) = lo g_{2} ((\frac{9}{20})^{\frac{9}{20}})

= - lo g_{2} ((\frac{9}{20})^{\frac{9}{20}}) - \frac{1}{20} lo g_{2} (\frac{1}{20})

対数の規則を適用する:

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

\frac{1}{20} lo g_{2} (\frac{1}{20}) = lo g_{2} ((\frac{1}{20})^{\frac{1}{20}})

= - lo g_{2} ((\frac{9}{20})^{\frac{9}{20}}) - lo g_{2} ((\frac{1}{20})^{\frac{1}{20}})

- lo g_{2} ((\frac{9}{20})^{\frac{9}{20}}) - lo g_{2} ((\frac{1}{20})^{\frac{1}{20}})

を書き換え

- (lo g_{2} ((\frac{9}{20})^{\frac{9}{20}}) + lo g_{2} ((\frac{1}{20})^{\frac{1}{20}}))

= - (lo g_{2} ((\frac{9}{20})^{\frac{9}{20}}) + lo g_{2} ((\frac{1}{20})^{\frac{1}{20}}))

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{9}{20})^{\frac{9}{20}}) + lo g_{2} ((\frac{1}{20})^{\frac{1}{20}}) = lo g_{2} ((\frac{9}{20})^{\frac{9}{20}} (\frac{1}{20})^{\frac{1}{20}})

= - lo g_{2} ((\frac{9}{20})^{\frac{9}{20}} (\frac{1}{20})^{\frac{1}{20}})

簡素化

lo g_{2} ((\frac{9}{20})^{\frac{9}{20}} (\frac{1}{20})^{\frac{1}{20}}) : \frac{1}{20} lo g_{2} ((\frac{9}{20})^{9} \frac{1}{20})

= - \frac{1}{20} lo g_{2} ((\frac{9}{20})^{9} \frac{1}{20})

(\frac{9}{20})^{9} = \frac{9 ^{9}}{2 0 ^{9}}

= - \frac{1}{20} lo g_{2} (\frac{1}{20} \cdot \frac{9 ^{9}}{2 0 ^{9}})

乗じる

\frac{9 ^{9}}{2 0 ^{9}} \cdot \frac{1}{20} : \frac{387420489}{2 0 ^{10}}

= - \frac{1}{20} lo g_{2} (\frac{387420489}{2 0 ^{10}})