簡約-(3/11)log_{2}(3/11)-(8/11)log_{2}(8/11)

解

簡約

- (\frac{3}{11}) lo g_{2} (\frac{3}{11}) - (\frac{8}{11}) lo g_{2} (\frac{8}{11})

- \frac{lo g _{2} ( \frac{27}{285311670611} ) + 24}{11}

十進法表記

0.84535 \dots

解答ステップ

- (\frac{3}{11}) lo g_{2} (\frac{3}{11}) - (\frac{8}{11}) lo g_{2} (\frac{8}{11})

規則を適用する:

(a) = a

(\frac{3}{11}) = \frac{3}{11}

= - \frac{3}{11} lo g_{2} (\frac{3}{11}) - \frac{8}{11} lo g_{2} (\frac{8}{11})

- \frac{3}{11} lo g_{2} (\frac{3}{11}) = - lo g_{2} ((\frac{3}{11})^{\frac{3}{11}})

- \frac{8}{11} lo g_{2} (\frac{8}{11}) = - lo g_{2} ((\frac{8}{11})^{\frac{8}{11}})

= - lo g_{2} ((\frac{3}{11})^{\frac{3}{11}}) - lo g_{2} ((\frac{8}{11})^{\frac{8}{11}})

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{8}{11})^{\frac{8}{11}}) - lo g_{2} ((\frac{3}{11})^{\frac{3}{11}}) = lo g_{2} ((\frac{8}{11})^{\frac{8}{11}} (\frac{3}{11})^{\frac{3}{11}})

= - lo g_{2} ((\frac{8}{11})^{\frac{8}{11}} (\frac{3}{11})^{\frac{3}{11}})

(\frac{8}{11})^{\frac{8}{11}} (\frac{3}{11})^{\frac{3}{11}} = \frac{4 11 4}{11 214358881} 11 \frac{27}{1331}

= - lo g_{2} (\frac{4 11 4}{11 214358881} 11 \frac{27}{1331})

対数の規則を適用する:

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

lo g_{2} (\frac{4 11 4}{11 214358881} 11 \frac{27}{1331}) = lo g_{2} (\frac{4 11 4}{11 214358881}) + lo g_{2} (11 \frac{27}{1331})

= - (lo g_{2} (\frac{4 11 4}{11 214358881}) + lo g_{2} (11 \frac{27}{1331}))

lo g_{2} (\frac{4 11 4}{11 214358881}) = \frac{24 - 11 lo g _{2} ( 11 214358881 )}{11}

= - (\frac{24 - 11 lo g _{2} ( 11 214358881 )}{11} + lo g_{2} (11 \frac{27}{1331}))

\frac{24 - 11 lo g _{2} ( 11 214358881 )}{11} + lo g_{2} (11 \frac{27}{1331}) = \frac{lo g _{2} ( 11 \frac{27}{1331} ) \cdot 11 + 24 - 11 lo g _{2} ( 11 214358881 )}{11}

= - \frac{lo g _{2} ( 11 \frac{27}{1331} ) \cdot 11 + 24 - 11 lo g _{2} ( 11 214358881 )}{11}

= - \frac{11 lo g _{2} ( 11 \frac{27}{1331} ) + 24 - 11 lo g _{2} ( 11 214358881 )}{11}

11 lo g_{2} (11 \frac{27}{1331}) + 24 - 11 lo g_{2} (11214358881) = lo g_{2} (\frac{27}{285311670611}) + 24

= - \frac{lo g _{2} ( \frac{27}{285311670611} ) + 24}{11}