簡約 2ln(3)+ln(x)-1/3 ln(x^4+1)

解

簡約

2 ln (3) + ln (x) - \frac{1}{3} ln (x^{4} + 1)

ln \frac{9 x ( x ^{4} + 1 ) ^{\frac{2}{3}}}{x ^{4} + 1}

解答ステップ

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

= ln (3^{2}) + ln (x) - \frac{1}{3} 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} (ab)

ln (3^{2}) + ln (x) = ln (3^{2} x)

= ln (3^{2} x) - \frac{1}{3} ln (x^{4} + 1)

3^{2} = 9

= ln (9 x) - \frac{1}{3} 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}{3} ln (x^{4} + 1) = ln ((x^{4} + 1)^{\frac{1}{3}})

= ln (9 x) - ln ((x^{4} + 1)^{\frac{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 (9 x) - ln ((x^{4} + 1)^{\frac{1}{3}}) = ln (\frac{9 x}{( x ^{4} + 1 ) ^{\frac{1}{3}}})

= ln (\frac{9 x}{( x ^{4} + 1 ) ^{\frac{1}{3}}})

簡素化

\frac{9 x}{( x ^{4} + 1 ) ^{\frac{1}{3}}} : \frac{9 x ( x ^{4} + 1 ) ^{\frac{2}{3}}}{x ^{4} + 1}

= ln \frac{9 x ( x ^{4} + 1 ) ^{\frac{2}{3}}}{x ^{4} + 1}