簡約-6(ln(3)-ln(2))+7/2-6ln(2)+15/2

解

簡約

- 6 (ln (3) - ln (2)) + \frac{7}{2} - 6 ln (2) + \frac{15}{2}

- 6 ln (3) + 11

十進法表記

4.40832 \dots

解答ステップ

- 6 (ln (3) - ln (2)) + \frac{7}{2} - 6 ln (2) + \frac{15}{2}

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

= - 6 ln (\frac{3}{2}) + \frac{7}{2} - 6 ln (2) + \frac{15}{2}

対数の規則を適用する:

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

6 ln (\frac{3}{2}) = ln ((\frac{3}{2})^{6})

= - ln ((\frac{3}{2})^{6}) + \frac{7}{2} - 6 ln (2) + \frac{15}{2}

対数の規則を適用する:

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

6 ln (2) = ln (2^{6})

= - ln ((\frac{3}{2})^{6}) + \frac{7}{2} - ln (2^{6}) + \frac{15}{2}

- ln ((\frac{3}{2})^{6}) - ln (2^{6})

を書き換え

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

= - (ln (2^{6}) + ln ((\frac{3}{2})^{6})) + \frac{7}{2} + \frac{15}{2}

対数の規則を適用する:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

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

= - ln (2^{6} (\frac{3}{2})^{6}) + \frac{7}{2} + \frac{15}{2}

(\frac{3}{2})^{6} \cdot 2^{6} = 729

= - ln (729) + \frac{7}{2} + \frac{15}{2}

分数を組み合わせる

\frac{7}{2} + \frac{15}{2} : 11

= - ln (729) + 11

ln (729) = 6 ln (3)

= - 6 ln (3) + 11