簡約 1/2 ln(2x)+5/4 ln(4x-4)-3/4 ln(4x-4)

解

簡約

\frac{1}{2} ln (2 x) + \frac{5}{4} ln (4 x - 4) - \frac{3}{4} ln (4 x - 4)

ln (22 x x - 1)

解答ステップ

\frac{1}{2} ln (2 x) + \frac{5}{4} ln (4 x - 4) - \frac{3}{4} ln (4 x - 4)

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

= ln ((2 x)^{\frac{1}{2}}) + \frac{5}{4} ln (4 x - 4) - \frac{3}{4} ln (4 x - 4)

対数の規則を適用する:

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

\frac{5}{4} ln (4 x - 4) = ln ((4 x - 4)^{\frac{5}{4}})

= ln ((2 x)^{\frac{1}{2}}) + ln ((4 x - 4)^{\frac{5}{4}}) - \frac{3}{4} ln (4 x - 4)

対数の規則を適用する:

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

ln ((2 x)^{\frac{1}{2}}) + ln ((4 x - 4)^{\frac{5}{4}}) = ln ((2 x)^{\frac{1}{2}} (4 x - 4)^{\frac{5}{4}})

= ln ((2 x)^{\frac{1}{2}} (4 x - 4)^{\frac{5}{4}}) - \frac{3}{4} ln (4 x - 4)

(2 x)^{\frac{1}{2}} = 2 x

= ln (2 x (4 x - 4)^{\frac{5}{4}}) - \frac{3}{4} ln (4 x - 4)

対数の規則を適用する:

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

\frac{3}{4} ln (4 x - 4) = ln ((4 x - 4)^{\frac{3}{4}})

= ln (2 x (4 x - 4)^{\frac{5}{4}}) - ln ((4 x - 4)^{\frac{3}{4}})

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 (4 x - 4)^{\frac{5}{4}}) - ln ((4 x - 4)^{\frac{3}{4}}) = ln (\frac{2 x ( 4 x - 4 ) ^{\frac{5}{4}}}{( 4 x - 4 ) ^{\frac{3}{4}}})

= ln (\frac{2 x ( 4 x - 4 ) ^{\frac{5}{4}}}{( 4 x - 4 ) ^{\frac{3}{4}}})

簡素化

\frac{2 x ( 4 x - 4 ) ^{\frac{5}{4}}}{( 4 x - 4 ) ^{\frac{3}{4}}} : 22 x x - 1

= ln (22 x x - 1)