簡約 4(1/9 ln|x-2|-1/9 ln|x+7|+C)

解

簡約

4 (\frac{1}{9} ln ∣ x - 2 ∣ - \frac{1}{9} ln ∣ x + 7 ∣ + C)

4 (ln (9 \frac{∣ x - 2 ∣}{∣ x + 7 ∣}) + C)

解答ステップ

4 (\frac{1}{9} ln ∣ x - 2 ∣ - \frac{1}{9} ln ∣ x + 7 ∣ + C)

対数の規則を適用する:

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

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

= 4 (ln (∣ x - 2 ∣^{\frac{1}{9}}) - \frac{1}{9} ln ∣ x + 7 ∣ + C)

対数の規則を適用する:

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

\frac{1}{9} ln ∣ x + 7 ∣ = ln (∣ x + 7 ∣^{\frac{1}{9}})

= 4 (ln (∣ x - 2 ∣^{\frac{1}{9}}) - ln (∣ x + 7 ∣^{\frac{1}{9}}) + C)

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 (∣ x - 2 ∣^{\frac{1}{9}}) - ln (∣ x + 7 ∣^{\frac{1}{9}}) = ln (\frac{∣ x - 2 ∣ ^{\frac{1}{9}}}{∣ x + 7 ∣ ^{\frac{1}{9}}})

= 4 (ln (\frac{∣ x - 2 ∣ ^{\frac{1}{9}}}{∣ x + 7 ∣ ^{\frac{1}{9}}}) + C)

同じべき乗を組み合わせる :

\frac{x ^{\frac{a}{m}}}{y ^{\frac{b}{m}}} = m \frac{x ^{a}}{y ^{b}}

= 4 (ln (9 \frac{∣ x - 2 ∣}{∣ x + 7 ∣}) + C)