簡約 (-ln(2/3)-1/3)/3+9ln(2/3)+3+3/2+1/9+1/36

解

簡約

\frac{- ln ( \frac{2}{3} ) - \frac{1}{3}}{3} + 9 ln (\frac{2}{3}) + 3 + \frac{3}{2} + \frac{1}{9} + \frac{1}{36}

9 (ln (2) - ln (3)) + \frac{163}{36} - \frac{ln ( \frac{2}{3} )}{3}

十進法表記

1.01374 \dots

解答ステップ

\frac{- ln ( \frac{2}{3} ) - \frac{1}{3}}{3} + 9 ln (\frac{2}{3}) + 3 + \frac{3}{2} + \frac{1}{9} + \frac{1}{36}

対数の規則を適用する:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

ln (\frac{2}{3}) = ln (2) - ln (3)

= \frac{- \frac{1}{3} - ln ( \frac{2}{3} )}{3} + 9 (ln (2) - ln (3)) + 3 + \frac{3}{2} + \frac{1}{9} + \frac{1}{36}

条件のようなグループ

= \frac{3}{2} + \frac{1}{9} + \frac{1}{36} + \frac{- \frac{1}{3} - ln ( \frac{2}{3} )}{3} + 9 (ln (2) - ln (3)) + 3

\frac{- \frac{1}{3} - ln ( \frac{2}{3} )}{3} = \frac{- 1 - 3 ln ( \frac{2}{3} )}{9}

= \frac{3}{2} + \frac{1}{9} + \frac{1}{36} + \frac{- 3 ln ( \frac{2}{3} ) - 1}{9} + 9 (ln (2) - ln (3)) + 3

分数を組み合わせる

\frac{1}{9} + \frac{- 3 ln ( \frac{2}{3} ) - 1}{9} : - \frac{ln ( \frac{2}{3} )}{3}

= \frac{3}{2} - \frac{ln ( \frac{2}{3} )}{3} + \frac{1}{36} + 9 (ln (2) - ln (3)) + 3

分数を組み合わせる

3 + \frac{3}{2} + \frac{1}{36} : \frac{163}{36}

= 9 (ln (2) - ln (3)) + \frac{163}{36} - \frac{ln ( \frac{2}{3} )}{3}