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

解

簡約

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

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

解答ステップ

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

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

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

対数の規則を適用する:

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

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

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

対数の規則を適用する:

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

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

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

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

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