簡約-8/14 log_{2}(8/14)-6/14 log_{2}(6/14)

解

簡約

- \frac{8}{14} lo g_{2} (\frac{8}{14}) - \frac{6}{14} lo g_{2} (\frac{6}{14})

- \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{8}{14} lo g_{2} (\frac{8}{14}) - \frac{6}{14} lo g_{2} (\frac{6}{14})

- \frac{8}{14} lo g_{2} (\frac{8}{14}) = - \frac{4}{7} lo g_{2} (\frac{4}{7})

\frac{6}{14} lo g_{2} (\frac{6}{14}) = \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})

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

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

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

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

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

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

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} (7 \frac{27}{343} \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}))

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