簡約-5/2 ln|x-2|+3/8 ln|x-3|+49/8 ln|x-1|+3

解

簡約

- \frac{5}{2} ln ∣ x - 2 ∣ + \frac{3}{8} ln ∣ x - 3 ∣ + \frac{49}{8} ln ∣ x - 1 ∣ + 3

ln (\frac{∣ x - 1 ∣ ^{\frac{49}{8}} ∣ x - 3 ∣ ^{\frac{3}{8}}}{∣ x - 2 ∣ ^{\frac{5}{2}}}) + 3

解答ステップ

- \frac{5}{2} ln ∣ x - 2 ∣ + \frac{3}{8} ln ∣ x - 3 ∣ + \frac{49}{8} ln ∣ x - 1 ∣ + 3

対数の規則を適用する:

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

\frac{5}{2} ln ∣ x - 2 ∣ = ln (∣ x - 2 ∣^{\frac{5}{2}})

= - ln (∣ x - 2 ∣^{\frac{5}{2}}) + \frac{3}{8} ln ∣ x - 3 ∣ + \frac{49}{8} ln ∣ x - 1 ∣ + 3

対数の規則を適用する:

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

\frac{3}{8} ln ∣ x - 3 ∣ = ln (∣ x - 3 ∣^{\frac{3}{8}})

= - ln (∣ x - 2 ∣^{\frac{5}{2}}) + ln (∣ x - 3 ∣^{\frac{3}{8}}) + \frac{49}{8} ln ∣ x - 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 (∣ x - 3 ∣^{\frac{3}{8}}) - ln (∣ x - 2 ∣^{\frac{5}{2}}) = ln (\frac{∣ x - 3 ∣ ^{\frac{3}{8}}}{∣ x - 2 ∣ ^{\frac{5}{2}}})

= ln (\frac{∣ x - 3 ∣ ^{\frac{3}{8}}}{∣ x - 2 ∣ ^{\frac{5}{2}}}) + \frac{49}{8} ln ∣ x - 1 ∣ + 3

対数の規則を適用する:

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

\frac{49}{8} ln ∣ x - 1 ∣ = ln (∣ x - 1 ∣^{\frac{49}{8}})

= ln (\frac{∣ x - 3 ∣ ^{\frac{3}{8}}}{∣ x - 2 ∣ ^{\frac{5}{2}}}) + ln (∣ x - 1 ∣^{\frac{49}{8}}) + 3

対数の規則を適用する:

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

ln (\frac{∣ x - 3 ∣ ^{\frac{3}{8}}}{∣ x - 2 ∣ ^{\frac{5}{2}}}) + ln (∣ x - 1 ∣^{\frac{49}{8}}) = ln (\frac{∣ x - 3 ∣ ^{\frac{3}{8}}}{∣ x - 2 ∣ ^{\frac{5}{2}}} ∣ x - 1 ∣^{\frac{49}{8}})

= ln (\frac{∣ x - 3 ∣ ^{\frac{3}{8}}}{∣ x - 2 ∣ ^{\frac{5}{2}}} ∣ x - 1 ∣^{\frac{49}{8}}) + 3

分数を乗じる:

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

= ln (\frac{∣ x - 1 ∣ ^{\frac{49}{8}} ∣ x - 3 ∣ ^{\frac{3}{8}}}{∣ x - 2 ∣ ^{\frac{5}{2}}}) + 3