簡約-3(1/2 ln(2^2-9)-1/2 ln(-2^2-9))

解

簡約

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

- \frac{3}{2} ln (\frac{5}{13})

十進法表記

1.43326 \dots

解答ステップ

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

対数の規則を適用する:

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

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

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

対数の規則を適用する:

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

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

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

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

\frac{( 2 ^{2} - 9 ) ^{\frac{1}{2}}}{( - 2 ^{2} - 9 ) ^{\frac{1}{2}}} = \frac{5}{13}

= - 3 ln (\frac{5}{13})

簡素化

ln (\frac{5}{13}) : \frac{1}{2} ln (\frac{5}{13})

= - 3 \cdot \frac{1}{2} ln (\frac{5}{13})

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{1 \cdot 3}{2} ln (\frac{5}{13})

数を乗じる:

1 \cdot 3 = 3

= - \frac{3}{2} ln (\frac{5}{13})