簡約 log_{sqrt(3)}(\sqrt[5]{1/81})

解

簡約

lo g_{3} (5 \frac{1}{81})

- \frac{8}{5}

十進法表記

- 1.6

解答ステップ

lo g_{3} (5 \frac{1}{81})

累乗根の規則を適用する:

a = a^{\frac{1}{2}}

3 = 3^{\frac{1}{2}}

= lo g_{3^{\frac{1}{2}}} (5 \frac{1}{81})

対数の規則を適用する:

lo g_{a^{b}} (x) = \frac{1}{b} lo g_{a} (x), a > 0

lo g_{3^{\frac{1}{2}}} (5 \frac{1}{81}) = \frac{1}{\frac{1}{2}} lo g_{3} (5 \frac{1}{81})

= \frac{1}{\frac{1}{2}} lo g_{3} (5 \frac{1}{81})

5 \frac{1}{81} = 3^{- \frac{4}{5}}

= \frac{1}{\frac{1}{2}} lo g_{3} (3^{- \frac{4}{5}})

対数の規則を適用する:

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

lo g_{3} (3^{- \frac{4}{5}}) = - \frac{4}{5}

= \frac{1}{\frac{1}{2}} (- \frac{4}{5})

\frac{1}{\frac{1}{2}} (- \frac{4}{5}) = - \frac{8}{5}

= - \frac{8}{5}