簡約 ln((x^5\sqrt[3]{x-2})/((x^2+1)cos(x)))

解

簡約

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

5 ln (x) + ln (3 x - 2) - ln (x^{2} + 1) - ln (cos (x))

解答ステップ

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

= ln (x^{5} 3 x - 2) - ln (cos (x) (x^{2} + 1))

対数の規則を適用する:

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

ln (x^{5} 3 x - 2) = ln (x^{5}) + ln (3 x - 2), ln ((x^{2} + 1) cos (x)) = ln (x^{2} + 1) + ln (cos (x))

= ln (x^{5}) + ln (3 x - 2) - (ln (x^{2} + 1) + ln (cos (x)))

対数の規則を適用する:

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

ln (x^{5}) = 5 ln (x)

= 5 ln (x) + ln (3 x - 2) - (ln (x^{2} + 1) + ln (cos (x)))

- (ln (x^{2} + 1) + ln (cos (x))) : - ln (x^{2} + 1) - ln (cos (x))

= 5 ln (x) + ln (3 x - 2) - ln (x^{2} + 1) - ln (cos (x))