簡約 ln(20)+2ln(1/2)+1/3 ln(27)

解

簡約

ln (20) + 2 ln (\frac{1}{2}) + \frac{1}{3} ln (27)

ln (15)

十進法表記

2.70805 \dots

解答ステップ

ln (20) + 2 ln (\frac{1}{2}) + \frac{1}{3} ln (27)

対数の規則を適用する:

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

2 ln (\frac{1}{2}) = ln ((\frac{1}{2})^{2})

= ln (20) + ln ((\frac{1}{2})^{2}) + \frac{1}{3} ln (27)

対数の規則を適用する:

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

ln (20) + ln ((\frac{1}{2})^{2}) = ln (20 (\frac{1}{2})^{2})

= ln (20 (\frac{1}{2})^{2}) + \frac{1}{3} ln (27)

20 (\frac{1}{2})^{2} = 5

= ln (5) + \frac{1}{3} ln (27)

対数の規則を適用する:

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

\frac{1}{3} ln (27) = ln (2 7^{\frac{1}{3}})

= ln (5) + ln (2 7^{\frac{1}{3}})

対数の規則を適用する:

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

ln (5) + ln (2 7^{\frac{1}{3}}) = ln (5 \cdot 2 7^{\frac{1}{3}})

= ln (5 \cdot 2 7^{\frac{1}{3}})

5 \cdot 2 7^{\frac{1}{3}} = 15

= ln (15)