簡約 ln(((4x^2+5)^3)/((3x-7)^2(5x^2-3x)^4))

解

簡約

ln (\frac{( 4 x ^{2} + 5 ) ^{3}}{( 3 x - 7 ) ^{2} ( 5 x ^{2} - 3 x ) ^{4}})

3 ln (4 x^{2} + 5) - 2 ln (3 x - 7) - 4 ln (5 x^{2} - 3 x)

解答ステップ

ln (\frac{( 4 x ^{2} + 5 ) ^{3}}{( 3 x - 7 ) ^{2} ( 5 x ^{2} - 3 x ) ^{4}})

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{( 4 x ^{2} + 5 ) ^{3}}{( 3 x - 7 ) ^{2} ( 5 x ^{2} - 3 x ) ^{4}}) = ln ((4 x^{2} + 5)^{3}) - ln ((3 x - 7)^{2} (5 x^{2} - 3 x)^{4})

= ln ((4 x^{2} + 5)^{3}) - ln ((3 x - 7)^{2} (5 x^{2} - 3 x)^{4})

対数の規則を適用する:

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

ln ((4 x^{2} + 5)^{3}) = 3 ln (4 x^{2} + 5)

= 3 ln (4 x^{2} + 5) - ln ((3 x - 7)^{2} (5 x^{2} - 3 x)^{4})

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 - 7)^{2} (5 x^{2} - 3 x)^{4}) = ln ((3 x - 7)^{2}) + ln ((5 x^{2} - 3 x)^{4})

= 3 ln (4 x^{2} + 5) - (ln ((3 x - 7)^{2}) + ln ((5 x^{2} - 3 x)^{4}))

対数の規則を適用する:

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

ln ((3 x - 7)^{2}) = 2 ln (3 x - 7)

= 3 ln (4 x^{2} + 5) - (2 ln (3 x - 7) + ln ((5 x^{2} - 3 x)^{4}))

対数の規則を適用する:

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

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

= 3 ln (4 x^{2} + 5) - (2 ln (3 x - 7) + 4 ln (5 x^{2} - 3 x))

- (2 ln (3 x - 7) + 4 ln (5 x^{2} - 3 x)) : - 2 ln (3 x - 7) - 4 ln (5 x^{2} - 3 x)

= 3 ln (4 x^{2} + 5) - 2 ln (3 x - 7) - 4 ln (5 x^{2} - 3 x)