簡約 ln(1+x^2)+1/10 ln(x)-ln(x+7)

解

簡約

ln (1 + x^{2}) + \frac{1}{10} ln (x) - ln (x + 7)

ln (\frac{( 1 + x ^{2} ) x ^{\frac{1}{10}}}{x + 7})

解答ステップ

ln (1 + x^{2}) + \frac{1}{10} ln (x) - ln (x + 7)

対数の規則を適用する:

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

\frac{1}{10} ln (x) = ln (x^{\frac{1}{10}})

= ln (x^{2} + 1) + ln (x^{\frac{1}{10}}) - ln (x + 7)

対数の規則を適用する:

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

ln (1 + x^{2}) + ln (x^{\frac{1}{10}}) = ln (x^{\frac{1}{10}} (x^{2} + 1))

= ln ((1 + x^{2}) x^{\frac{1}{10}}) - ln (x + 7)

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^{\frac{1}{10}} (1 + x^{2})) - ln (x + 7) = ln (\frac{x ^{\frac{1}{10}} ( x ^{2} + 1 )}{x + 7})

= ln (\frac{( 1 + x ^{2} ) x ^{\frac{1}{10}}}{x + 7})