簡約 1/3 ln(8)-2ln(2)-1/3 ln(5)

解

簡約

\frac{1}{3} ln (8) - 2 ln (2) - \frac{1}{3} ln (5)

ln (\frac{5 ^{\frac{2}{3}}}{10})

十進法表記

- 1.22962 \dots

解答ステップ

\frac{1}{3} ln (8) - 2 ln (2) - \frac{1}{3} 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{1}{3} ln (8) - ln (2^{2}) - \frac{1}{3} ln (5)

対数の規則を適用する:

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

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

= \frac{1}{3} ln (8) - ln (2^{2}) - ln (5^{\frac{1}{3}})

- ln (2^{2}) - ln (5^{\frac{1}{3}})

を書き換え

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

= \frac{1}{3} ln (8) - (ln (2^{2}) + ln (5^{\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 (2^{2}) + ln (5^{\frac{1}{3}}) = ln (2^{2} \cdot 5^{\frac{1}{3}})

= \frac{1}{3} ln (8) - ln (2^{2} \cdot 5^{\frac{1}{3}})

2^{2} = 4

= \frac{1}{3} ln (8) - ln (4 \cdot 5^{\frac{1}{3}})

対数の規則を適用する:

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

\frac{1}{3} ln (8) = ln (8^{\frac{1}{3}})

= ln (8^{\frac{1}{3}}) - ln (4 \cdot 5^{\frac{1}{3}})

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 (8^{\frac{1}{3}}) - ln (4 \cdot 5^{\frac{1}{3}}) = ln (\frac{8 ^{\frac{1}{3}}}{4 \cdot 5 ^{\frac{1}{3}}})

= ln (\frac{8 ^{\frac{1}{3}}}{4 \cdot 5 ^{\frac{1}{3}}})

\frac{8 ^{\frac{1}{3}}}{4 \cdot 5 ^{\frac{1}{3}}} = \frac{5 ^{\frac{2}{3}}}{10}

= ln (\frac{5 ^{\frac{2}{3}}}{10})