簡約 3/2 (ln(x^2-1)-ln(x+1)+ln(x-1))

解

簡約

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

3 ln (x - 1)

解答ステップ

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

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

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

簡素化

\frac{x ^{2} - 1}{x + 1} : x - 1

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

分数を乗じる:

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

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

3 (ln (x - 1) + ln (x - 1)) = 6 ln (x - 1)

= \frac{6 ln ( x - 1 )}{2}

数を割る:

\frac{6}{2} = 3

= 3 ln (x - 1)