簡約 1/3 ln(x)+10/12 ln(x+3)-1/6 ln(x-3)

解

簡約

\frac{1}{3} ln (x) + \frac{10}{12} ln (x + 3) - \frac{1}{6} ln (x - 3)

ln (\frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}} ( x - 3 ) ^{\frac{5}{6}}}{x - 3})

解答ステップ

\frac{1}{3} ln (x) + \frac{10}{12} ln (x + 3) - \frac{1}{6} ln (x - 3)

対数の規則を適用する:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{3} ln (x) = ln (x^{\frac{1}{3}})

= ln (x^{\frac{1}{3}}) + \frac{10}{12} ln (x + 3) - \frac{1}{6} ln (x - 3)

対数の規則を適用する:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{10}{12} ln (x + 3) = ln ((x + 3)^{\frac{10}{12}})

= ln (x^{\frac{1}{3}}) + ln ((x + 3)^{\frac{10}{12}}) - \frac{1}{6} ln (x - 3)

対数の規則を適用する:

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

ln (x^{\frac{1}{3}}) + ln ((x + 3)^{\frac{10}{12}}) = ln (x^{\frac{1}{3}} (x + 3)^{\frac{10}{12}})

= ln (x^{\frac{1}{3}} (x + 3)^{\frac{10}{12}}) - \frac{1}{6} ln (x - 3)

(x + 3)^{\frac{10}{12}} = (x + 3)^{\frac{5}{6}}

= ln (x^{\frac{1}{3}} (x + 3)^{\frac{5}{6}}) - \frac{1}{6} ln (x - 3)

対数の規則を適用する:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{6} ln (x - 3) = ln ((x - 3)^{\frac{1}{6}})

= ln (x^{\frac{1}{3}} (x + 3)^{\frac{5}{6}}) - ln ((x - 3)^{\frac{1}{6}})

対数の規則を適用する:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (x^{\frac{1}{3}} (x + 3)^{\frac{5}{6}}) - ln ((x - 3)^{\frac{1}{6}}) = ln (\frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}}}{( x - 3 ) ^{\frac{1}{6}}})

= ln (\frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}}}{( x - 3 ) ^{\frac{1}{6}}})

簡素化

\frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}}}{( x - 3 ) ^{\frac{1}{6}}} : \frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}} ( x - 3 ) ^{\frac{5}{6}}}{x - 3}

= ln (\frac{x ^{\frac{1}{3}} ( x + 3 ) ^{\frac{5}{6}} ( x - 3 ) ^{\frac{5}{6}}}{x - 3})