化简-[ 1/5 log_{2}(1/5)+4/5 log_{2}(4/5)]

解答

化简

- [\frac{1}{5} lo g_{2} (\frac{1}{5}) + \frac{4}{5} lo g_{2} (\frac{4}{5})]

- lo g_{2} (\frac{2 5 8}{5 625 5 5})

十进制

0.72192 \dots

求解步骤

- (\frac{1}{5} lo g_{2} (\frac{1}{5}) + \frac{4}{5} lo g_{2} (\frac{4}{5}))

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

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

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

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

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

使用对数计算法则:

lo g_{a} (x) - lo g_{a} (y) = lo g_{a} (\frac{x}{y})

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

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

\frac{( \frac{4}{5} ) ^{\frac{4}{5}}}{5 ^{\frac{1}{5}}} = \frac{2 5 8}{5 625 \cdot 5 ^{\frac{1}{5}}}

= - lo g_{2} (\frac{2 5 8}{5 625 \cdot 5 ^{\frac{1}{5}}})

使用指数法则:

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

5^{\frac{1}{5}} = 55

= - lo g_{2} (\frac{2 5 8}{5 625 5 5})

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