簡約 1/(2(x-1))+5/8 ln|x-1|-5/8 ln|x+3|

解

簡約

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

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

解答ステップ

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

対数の規則を適用する:

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

\frac{5}{8} ln ∣ x - 1 ∣ = ln (∣ x - 1 ∣^{\frac{5}{8}})

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

対数の規則を適用する:

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

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

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

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

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

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}}

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