簡約 1/2 ln(2x+5)-1/3 ln(x+8)+ln(C)

解

簡約

\frac{1}{2} ln (2 x + 5) - \frac{1}{3} ln (x + 8) + ln (C)

ln (\frac{C ( 2 x + 5 ) ^{\frac{1}{2}} ( x + 8 ) ^{\frac{2}{3}}}{x + 8})

解答ステップ

\frac{1}{2} ln (2 x + 5) - \frac{1}{3} ln (x + 8) + ln (C)

対数の規則を適用する:

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

\frac{1}{2} ln (2 x + 5) = ln ((2 x + 5)^{\frac{1}{2}})

= ln ((2 x + 5)^{\frac{1}{2}}) - \frac{1}{3} ln (x + 8) + ln (C)

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 + 8) = ln ((x + 8)^{\frac{1}{3}})

= ln ((2 x + 5)^{\frac{1}{2}}) - ln ((x + 8)^{\frac{1}{3}}) + ln (C)

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 ((2 x + 5)^{\frac{1}{2}}) - ln ((x + 8)^{\frac{1}{3}}) = ln (\frac{( 2 x + 5 ) ^{\frac{1}{2}}}{( x + 8 ) ^{\frac{1}{3}}})

= ln (\frac{( 2 x + 5 ) ^{\frac{1}{2}}}{( x + 8 ) ^{\frac{1}{3}}}) + ln (C)

簡素化

\frac{( 2 x + 5 ) ^{\frac{1}{2}}}{( x + 8 ) ^{\frac{1}{3}}} : \frac{( 2 x + 5 ) ^{\frac{1}{2}} ( x + 8 ) ^{\frac{2}{3}}}{x + 8}

= ln (\frac{( 2 x + 5 ) ^{\frac{1}{2}} ( x + 8 ) ^{\frac{2}{3}}}{x + 8}) + ln (C)

対数の規則を適用する:

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

ln (\frac{( 2 x + 5 ) ^{\frac{1}{2}} ( x + 8 ) ^{\frac{2}{3}}}{x + 8}) + ln (C) = ln (C \frac{( x + 8 ) ^{\frac{2}{3}} ( 2 x + 5 ) ^{\frac{1}{2}}}{x + 8})

= ln (C \frac{( 2 x + 5 ) ^{\frac{1}{2}} ( x + 8 ) ^{\frac{2}{3}}}{x + 8})

簡素化

\frac{( 2 x + 5 ) ^{\frac{1}{2}} ( x + 8 ) ^{\frac{2}{3}}}{x + 8} C : \frac{C ( 2 x + 5 ) ^{\frac{1}{2}} ( x + 8 ) ^{\frac{2}{3}}}{x + 8}

= ln (\frac{C ( 2 x + 5 ) ^{\frac{1}{2}} ( x + 8 ) ^{\frac{2}{3}}}{x + 8})