簡約-(3/7)log_{2}(3/7)-(4/7)log_{2}(4/7)

解

簡約

- (\frac{3}{7}) lo g_{2} (\frac{3}{7}) - (\frac{4}{7}) lo g_{2} (\frac{4}{7})

- \frac{lo g _{2} ( \frac{27}{7 ^{7}} ) + 8}{7}

十進法表記

0.98522 \dots

解答ステップ

- (\frac{3}{7}) lo g_{2} (\frac{3}{7}) - (\frac{4}{7}) lo g_{2} (\frac{4}{7})

規則を適用する:

(a) = a

(\frac{3}{7}) = \frac{3}{7}

= - \frac{3}{7} lo g_{2} (\frac{3}{7}) - \frac{4}{7} lo g_{2} (\frac{4}{7})

- \frac{3}{7} lo g_{2} (\frac{3}{7}) = - lo g_{2} ((\frac{3}{7})^{\frac{3}{7}})

- \frac{4}{7} lo g_{2} (\frac{4}{7}) = - lo g_{2} ((\frac{4}{7})^{\frac{4}{7}})

= - lo g_{2} ((\frac{3}{7})^{\frac{3}{7}}) - lo g_{2} ((\frac{4}{7})^{\frac{4}{7}})

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{4}{7})^{\frac{4}{7}}) - lo g_{2} ((\frac{3}{7})^{\frac{3}{7}}) = lo g_{2} ((\frac{4}{7})^{\frac{4}{7}} (\frac{3}{7})^{\frac{3}{7}})

= - lo g_{2} ((\frac{4}{7})^{\frac{4}{7}} (\frac{3}{7})^{\frac{3}{7}})

(\frac{4}{7})^{\frac{4}{7}} (\frac{3}{7})^{\frac{3}{7}} = \frac{2 7 2}{7 2401} 7 \frac{27}{343}

= - lo g_{2} (\frac{2 7 2}{7 2401} 7 \frac{27}{343})

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{2 7 2}{7 2401} 7 \frac{27}{343}) = lo g_{2} (\frac{2 7 2}{7 2401}) + lo g_{2} (7 \frac{27}{343})

= - (lo g_{2} (\frac{2 7 2}{7 2401}) + lo g_{2} (7 \frac{27}{343}))

lo g_{2} (\frac{2 7 2}{7 2401}) = \frac{8 - 7 lo g _{2} ( 7 2401 )}{7}

= - (\frac{8 - 7 lo g _{2} ( 7 2401 )}{7} + lo g_{2} (7 \frac{27}{343}))

\frac{8 - 7 lo g _{2} ( 7 2401 )}{7} + lo g_{2} (7 \frac{27}{343}) = \frac{lo g _{2} ( 7 \frac{27}{343} ) \cdot 7 + 8 - 7 lo g _{2} ( 7 2401 )}{7}

= - \frac{lo g _{2} ( 7 \frac{27}{343} ) \cdot 7 + 8 - 7 lo g _{2} ( 7 2401 )}{7}

= - \frac{7 lo g _{2} ( 7 \frac{27}{343} ) + 8 - 7 lo g _{2} ( 7 2401 )}{7}

7 lo g_{2} (7 \frac{27}{343}) + 8 - 7 lo g_{2} (72401) = lo g_{2} (\frac{27}{7 ^{7}}) + 8

= - \frac{lo g _{2} ( \frac{27}{7 ^{7}} ) + 8}{7}