簡約 ln(x)-2ln(x^2+1)+1/2 ln(x^4+1)

解

簡約

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

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

解答ステップ

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

対数の規則を適用する:

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

2 ln (x^{2} + 1) = ln ((x^{2} + 1)^{2})

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

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

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

対数の規則を適用する:

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

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

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

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

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

分数を乗じる:

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

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