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受欢迎的代数 >

化简 3/5*(-2/3 log_{e}(2/3)-1/3 log_{e}(1/3))

解答

化简

\frac{3}{5} \cdot (- \frac{2}{3} lo g_{e} (\frac{2}{3}) - \frac{1}{3} lo g_{e} (\frac{1}{3}))

解答

- \frac{ln ( \frac{4}{27} )}{5}

+1

十进制

0.38190 \dots

求解步骤

\frac{3}{5} (- \frac{2}{3} lo g_{e} (\frac{2}{3}) - \frac{1}{3} lo g_{e} (\frac{1}{3}))

使用对数计算法则:

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

\frac{2}{3} lo g_{e} (\frac{2}{3}) = lo g_{e} ((\frac{2}{3})^{\frac{2}{3}})

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

使用对数计算法则:

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

\frac{1}{3} lo g_{e} (\frac{1}{3}) = lo g_{e} ((\frac{1}{3})^{\frac{1}{3}})

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

将

- lo g_{e} ((\frac{2}{3})^{\frac{2}{3}}) - lo g_{e} ((\frac{1}{3})^{\frac{1}{3}})

改写为

- (lo g_{e} ((\frac{2}{3})^{\frac{2}{3}}) + lo g_{e} ((\frac{1}{3})^{\frac{1}{3}}))

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

使用对数计算法则:

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

lo g_{e} ((\frac{2}{3})^{\frac{2}{3}}) + lo g_{e} ((\frac{1}{3})^{\frac{1}{3}}) = lo g_{e} ((\frac{2}{3})^{\frac{2}{3}} (\frac{1}{3})^{\frac{1}{3}})

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

去除括号:

(- a) = - a

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

化简

lo g_{e} ((\frac{2}{3})^{\frac{2}{3}} (\frac{1}{3})^{\frac{1}{3}}) : ln ((\frac{2}{3})^{\frac{2}{3}} (\frac{1}{3})^{\frac{1}{3}})

= - \frac{3}{5} ln ((\frac{2}{3})^{\frac{2}{3}} (\frac{1}{3})^{\frac{1}{3}})

化简

ln ((\frac{2}{3})^{\frac{2}{3}} (\frac{1}{3})^{\frac{1}{3}}) : \frac{1}{3} ln ((\frac{2}{3})^{2} \frac{1}{3})

= - \frac{3}{5} \cdot \frac{1}{3} ln ((\frac{2}{3})^{2} \frac{1}{3})

分式相乘:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

= - \frac{3 \cdot 1 \cdot ln ( ( \frac{2}{3} ) ^{2} \frac{1}{3} )}{5 \cdot 3}

约分:

3

= - \frac{1 \cdot ln ( ( \frac{2}{3} ) ^{2} \frac{1}{3} )}{5}

乘以:

1 \cdot ln ((\frac{2}{3})^{2} \frac{1}{3}) = ln ((\frac{2}{3})^{2} \frac{1}{3})

= - \frac{ln ( ( \frac{2}{3} ) ^{2} \frac{1}{3} )}{5}

(\frac{2}{3})^{2} = \frac{2 ^{2}}{3 ^{2}}

= - \frac{ln ( \frac{1}{3} \cdot \frac{2 ^{2}}{3 ^{2}} )}{5}

乘

\frac{2 ^{2}}{3 ^{2}} \cdot \frac{1}{3} : \frac{4}{27}

= - \frac{ln ( \frac{4}{27} )}{5}

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