簡約 54(1/5 ln|n+1|-1/5 (ln|n+6|-ln(6)))

解

簡約

54 (\frac{1}{5} ln ∣ n + 1 ∣ - \frac{1}{5} (ln ∣ n + 6 ∣ - ln (6)))

54 ln (\frac{5 6 ∣ n + 1 ∣ ^{\frac{1}{5}} ∣ n + 6 ∣ ^{\frac{4}{5}}}{∣ n + 6 ∣})

解答ステップ

54 (\frac{1}{5} ln ∣ n + 1 ∣ - \frac{1}{5} (ln ∣ n + 6 ∣ - ln (6)))

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 ∣ n + 6 ∣ - ln (6) = ln (\frac{∣ n + 6 ∣}{6})

= 54 (\frac{1}{5} ln ∣ n + 1 ∣ - \frac{1}{5} ln (\frac{∣ n + 6 ∣}{6}))

対数の規則を適用する:

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

\frac{1}{5} ln ∣ n + 1 ∣ = ln (∣ n + 1 ∣^{\frac{1}{5}})

= 54 (ln (∣ n + 1 ∣^{\frac{1}{5}}) - \frac{1}{5} ln (\frac{∣ n + 6 ∣}{6}))

対数の規則を適用する:

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

\frac{1}{5} ln (\frac{∣ n + 6 ∣}{6}) = ln ((\frac{∣ n + 6 ∣}{6})^{\frac{1}{5}})

= 54 (ln (∣ n + 1 ∣^{\frac{1}{5}}) - ln ((\frac{∣ n + 6 ∣}{6})^{\frac{1}{5}}))

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 (∣ n + 1 ∣^{\frac{1}{5}}) - ln ((\frac{∣ n + 6 ∣}{6})^{\frac{1}{5}}) = ln \frac{∣ n + 1 ∣ ^{\frac{1}{5}}}{( \frac{∣ n + 6 ∣}{6} ) ^{\frac{1}{5}}}

= 54 ln \frac{∣ n + 1 ∣ ^{\frac{1}{5}}}{( \frac{∣ n + 6 ∣}{6} ) ^{\frac{1}{5}}}

簡素化

\frac{∣ n + 1 ∣ ^{\frac{1}{5}}}{( \frac{∣ n + 6 ∣}{6} ) ^{\frac{1}{5}}} : \frac{5 6 ∣ n + 1 ∣ ^{\frac{1}{5}} ∣ n + 6 ∣ ^{\frac{4}{5}}}{∣ n + 6 ∣}

= 54 ln (\frac{5 6 ∣ n + 1 ∣ ^{\frac{1}{5}} ∣ n + 6 ∣ ^{\frac{4}{5}}}{∣ n + 6 ∣})