化简 log_{sqrt(3)}(\sqrt[5]{1/81})

解答

化简

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

- \frac{8}{5}

十进制

- 1.6

求解步骤

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

使用根式运算法则:

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

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

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

使用对数计算法则:

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}})

使用对数计算法则:

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}

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