簡約-1/6 ln|5-6(7)|-3ln|7|

解

簡約

- \frac{1}{6} ln ∣ 5 - 6 (7) ∣ - 3 ln ∣ 7 ∣

- ln (343 \cdot 3 7^{\frac{1}{6}})

十進法表記

- 6.43955 \dots

解答ステップ

- \frac{1}{6} ln ∣ 5 - 6 (7) ∣ - 3 ln ∣ 7 ∣

対数の規則を適用する:

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

\frac{1}{6} ln ∣ 5 - 6 (7) ∣ = ln (∣ 5 - 6 (7) ∣^{\frac{1}{6}})

= - ln (∣ 5 - 6 (7) ∣^{\frac{1}{6}}) - 3 ln ∣ 7 ∣

対数の規則を適用する:

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

3 ln ∣ 7 ∣ = ln (∣ 7 ∣^{3})

= - ln (∣ 5 - 6 (7) ∣^{\frac{1}{6}}) - ln (∣ 7 ∣^{3})

- ln (∣ 5 - 6 (7) ∣^{\frac{1}{6}}) - ln (∣ 7 ∣^{3})

を書き換え

- (ln (∣ 5 - 6 (7) ∣^{\frac{1}{6}}) + ln (∣ 7 ∣^{3}))

= - (ln (∣ 7 ∣^{3}) + ln (∣ 5 - 6 (7) ∣^{\frac{1}{6}}))

対数の規則を適用する:

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

ln (∣ 5 - 6 (7) ∣^{\frac{1}{6}}) + ln (∣ 7 ∣^{3}) = ln (∣ 7 ∣^{3} ∣ 5 - 6 \cdot 7 ∣^{\frac{1}{6}})

= - ln (∣ 7 ∣^{3} ∣ 5 - 6 \cdot 7 ∣^{\frac{1}{6}})

∣ 5 - 6 (7) ∣^{\frac{1}{6}} ∣ 7 ∣^{3} = 343 \cdot 3 7^{\frac{1}{6}}

= - ln (343 \cdot 3 7^{\frac{1}{6}})