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

解

簡約

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

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

解答ステップ

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

対数の規則を適用する:

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

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

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

対数の規則を適用する:

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

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

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

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

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

簡素化

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

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