簡約 1/3 ln(x)+10/12 (x+3)-1/6 ln(x-3)

解

簡約

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

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

解答ステップ

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

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

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

簡素化

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

= ln (\frac{x ^{\frac{1}{3}} ( x - 3 ) ^{\frac{5}{6}}}{x - 3}) + \frac{10}{12} (x + 3)

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

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