簡約 ln((e^{(x+2)}(5x+1)^3)/(x^5(2x+1)^7))

解

簡約

ln (\frac{e ^{(x + 2)} ( 5 x + 1 ) ^{3}}{x ^{5} ( 2 x + 1 ) ^{7}})

x + 2 + 3 ln (5 x + 1) - 5 ln (x) - 7 ln (2 x + 1)

解答ステップ

ln (\frac{e ^{(x + 2)} ( 5 x + 1 ) ^{3}}{x ^{5} ( 2 x + 1 ) ^{7}})

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{e ^{(x + 2)} ( 5 x + 1 ) ^{3}}{x ^{5} ( 2 x + 1 ) ^{7}}) = ln (e^{(x + 2)} (5 x + 1)^{3}) - ln (x^{5} (2 x + 1)^{7})

= ln (e^{x + 2} (5 x + 1)^{3}) - ln (x^{5} (2 x + 1)^{7})

対数の規則を適用する:

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

ln (e^{x + 2} (5 x + 1)^{3}) = ln (e^{x + 2}) + ln ((5 x + 1)^{3}), ln (x^{5} (2 x + 1)^{7}) = ln (x^{5}) + ln ((2 x + 1)^{7})

= ln (e^{x + 2}) + ln ((5 x + 1)^{3}) - (ln (x^{5}) + ln ((2 x + 1)^{7}))

対数の規則を適用する:

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

ln (e^{x + 2}) = (x + 2) ln (e)

= (x + 2) ln (e) + ln ((5 x + 1)^{3}) - (ln (x^{5}) + ln ((2 x + 1)^{7}))

対数の規則を適用する:

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

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

= (x + 2) ln (e) + 3 ln (5 x + 1) - (ln (x^{5}) + ln ((2 x + 1)^{7}))

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)

= (x + 2) ln (e) + 3 ln (5 x + 1) - (5 ln (x) + ln ((2 x + 1)^{7}))

対数の規則を適用する:

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

ln ((2 x + 1)^{7}) = 7 ln (2 x + 1)

= (x + 2) ln (e) + 3 ln (5 x + 1) - (5 ln (x) + 7 ln (2 x + 1))

(x + 2) ln (e) = x + 2

= x + 2 + 3 ln (5 x + 1) - (5 ln (x) + 7 ln (2 x + 1))

- (5 ln (x) + 7 ln (2 x + 1)) : - 5 ln (x) - 7 ln (2 x + 1)

= x + 2 + 3 ln (5 x + 1) - 5 ln (x) - 7 ln (2 x + 1)