簡約 7/6+2ln(2)-ln(3)-5(ln(6)-ln(5))

解

簡約

\frac{7}{6} + 2 ln (2) - ln (3) - 5 (ln (6) - ln (5))

\frac{7}{6} + ln (\frac{3125}{5832})

十進法表記

0.54274 \dots

解答ステップ

\frac{7}{6} + 2 ln (2) - ln (3) - 5 (ln (6) - ln (5))

対数の規則を適用する:

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

2 ln (2) = ln (2^{2})

= \frac{7}{6} + ln (2^{2}) - ln (3) - 5 (ln (6) - ln (5))

対数の規則を適用する:

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

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

= \frac{7}{6} + ln (\frac{2 ^{2}}{3}) - 5 (ln (6) - ln (5))

2^{2} = 4

= \frac{7}{6} + ln (\frac{4}{3}) - 5 (ln (6) - ln (5))

対数の規則を適用する:

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

ln (6) - ln (5) = ln (\frac{6}{5})

= \frac{7}{6} + ln (\frac{4}{3}) - 5 ln (\frac{6}{5})

対数の規則を適用する:

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

5 ln (\frac{6}{5}) = ln ((\frac{6}{5})^{5})

= \frac{7}{6} + ln (\frac{4}{3}) - ln ((\frac{6}{5})^{5})

対数の規則を適用する:

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

ln (\frac{4}{3}) - ln ((\frac{6}{5})^{5}) = ln (\frac{\frac{4}{3}}{( \frac{6}{5} ) ^{5}})

= \frac{7}{6} + ln (\frac{\frac{4}{3}}{( \frac{6}{5} ) ^{5}})

\frac{\frac{4}{3}}{( \frac{6}{5} ) ^{5}} = \frac{3125}{5832}

= \frac{7}{6} + ln (\frac{3125}{5832})