簡約 3ln(2)+2ln(x)-1/2 ln(x+3)

解

簡約

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

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

解答ステップ

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

対数の規則を適用する:

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

3 ln (2) = ln (2^{3})

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

対数の規則を適用する:

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

2 ln (x) = ln (x^{2})

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

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

2^{3} = 8

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

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

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

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

簡素化

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

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