簡約 ln(1/x (1/x)^{3/2})

解

簡約

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

- \frac{5}{2} ln (x)

解答ステップ

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

対数の規則を適用する:

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

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

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

対数の規則を適用する:

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

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

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

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{1}{x}) = ln (1) - ln (x), ln (\frac{1}{x}) = ln (1) - ln (x)

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

ln (1) = 0

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

= 0 - ln (x) - \frac{3}{2} ln (x)

類似した元を足す:

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

= 0 - \frac{5}{2} ln (x)

0 - \frac{5}{2} ln (x) = - \frac{5}{2} ln (x)

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