簡約 2ln(x+1)+1/3 ln(x)-ln(cos(x))

解

簡約

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

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

解答ステップ

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

対数の規則を適用する:

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

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

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

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

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

対数の規則を適用する:

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

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

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

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

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

基本的な三角関数の公式を使用する:

\frac{1}{cos ( x )} = sec (x)

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