簡約 ln((x^5)/(sqrt(5)(1+x)^4))

解

簡約

ln (\frac{x ^{5}}{5 ( 1 + x ) ^{4}})

5 ln (x) - \frac{1}{2} ln (5) - 4 ln (1 + x)

解答ステップ

ln (\frac{x ^{5}}{5 ( 1 + x ) ^{4}})

対数の規則を適用する:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

ln (\frac{x ^{5}}{5 ( 1 + x ) ^{4}}) = ln (x^{5}) - ln (5 (1 + x)^{4})

= ln (x^{5}) - ln (5 (x + 1)^{4})

対数の規則を適用する:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (x^{5}) = 5 ln (x)

= 5 ln (x) - ln (5 (1 + x)^{4})

対数の規則を適用する:

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

ln (5 (1 + x)^{4}) = ln (5) + ln ((1 + x)^{4})

= 5 ln (x) - (ln ((x + 1)^{4}) + ln (5))

対数の規則を適用する:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln ((1 + x)^{4}) = 4 ln (1 + x)

= 5 ln (x) - (ln (5) + 4 ln (1 + x))

ln (5) = \frac{1}{2} ln (5)

= 5 ln (x) - (4 ln (x + 1) + \frac{1}{2} ln (5))

- (\frac{1}{2} ln (5) + 4 ln (1 + x)) : - \frac{1}{2} ln (5) - 4 ln (1 + x)

= 5 ln (x) - \frac{1}{2} ln (5) - 4 ln (1 + x)