簡約 ln((\sqrt[3]{a^2b^4})/(2c^5))

解

簡約

ln (\frac{3 a ^{2} b ^{4}}{2 c ^{5}})

ln (3 b 3 b^{3} 3 a^{2}) - ln (2) - 5 ln (c)

解答ステップ

ln (\frac{3 a ^{2} b ^{4}}{2 c ^{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{3 a ^{2} b ^{4}}{2 c ^{5}}) = ln (3 a^{2} b^{4}) - ln (2 c^{5})

= ln (3 a^{2} b^{4}) - ln (2 c^{5})

対数の規則を適用する:

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

ln (2 c^{5}) = ln (2) + ln (c^{5})

= ln (3 a^{2} b^{4}) - (ln (c^{5}) + ln (2))

対数の規則を適用する:

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

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

= ln (3 a^{2} b^{4}) - (ln (2) + 5 ln (c))

3 a^{2} b^{4} = 3 b 3 b^{3} 3 a^{2}

= ln (3 b 3 b^{3} 3 a^{2}) - (5 ln (c) + ln (2))

= ln (3 b 3 b^{3} 3 a^{2}) - (ln (2) + 5 ln (c))

- (ln (2) + 5 ln (c)) : - ln (2) - 5 ln (c)

= ln (3 b^{3} 3 b 3 a^{2}) - ln (2) - 5 ln (c)

= ln (3 b 3 b^{3} 3 a^{2}) - ln (2) - 5 ln (c)