簡約 ln(x^2-9)+2ln(1/(x+3))+4ln(x)

解

簡約

ln (x^{2} - 9) + 2 ln (\frac{1}{x + 3}) + 4 ln (x)

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

解答ステップ

ln (x^{2} - 9) + 2 ln (\frac{1}{x + 3}) + 4 ln (x)

対数の規則を適用する:

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

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

= ln (x^{2} - 9) + ln ((\frac{1}{x + 3})^{2}) + 4 ln (x)

対数の規則を適用する:

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

ln (x^{2} - 9) + ln ((\frac{1}{x + 3})^{2}) = ln ((\frac{1}{x + 3})^{2} (x^{2} - 9))

= ln ((\frac{1}{x + 3})^{2} (x^{2} - 9)) + 4 ln (x)

簡素化

(x^{2} - 9) (\frac{1}{x + 3})^{2} : \frac{x - 3}{x + 3}

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

対数の規則を適用する:

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

4 ln (x) = ln (x^{4})

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

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

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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