展开 ln(\sqrt[3]{(x+2)/(x^4(x^2-4))})

解答

展开

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

- \frac{1}{3} ln (x^{5} - 2 x^{4})

求解步骤

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

3 \frac{x + 2}{x ^{4} ( x ^{2} - 4 )} = 3 \frac{1}{x ^{5} - 2 x ^{4}}

= ln (3 \frac{1}{x ^{5} - 2 x ^{4}})

改写为

= ln ((\frac{1}{x ^{5} - 2 x ^{4}})^{\frac{1}{3}})

使用对数运算法则:

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

假定

x \geq 0

= \frac{1}{3} ln (\frac{1}{x ^{5} - 2 x ^{4}})

化简

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

= \frac{1}{3} (- ln (x^{5} - 2 x^{4}))

去除括号:

(- a) = - a

= - \frac{1}{3} ln (x^{5} - 2 x^{4})

e x p an d a^{6} - b^{6} - 3 a^{2} b^{2} (a + b) (a - b)

e x p an d \frac{( 2 x - 8 ) ^{2}}{9}

e x p an d \frac{1}{2} + (x - \frac{1}{2}) e^{x}

e x p an d \frac{1 + 3 x}{( 1 + 2 x )}

e x p an d \frac{1}{2} p o w (x, 4) - p o w (x + 1, 3) - 3 p o w (x, 2)