化简 1/4 log_{10}(0.485)+1/3 log_{10}(0.36)

解答

化简

\frac{1}{4} lo g_{10} (0.485) + \frac{1}{3} lo g_{10} (0.36)

lo g_{10} (4 \frac{97}{200} 3 \frac{9}{25})

十进制

- 0.22646 \dots

求解步骤

\frac{1}{4} lo g_{10} (0.485) + \frac{1}{3} lo g_{10} (0.36)

\frac{1}{4} lo g_{10} (0.485) = \frac{1}{4} lo g_{10} (\frac{97}{200})

\frac{1}{3} lo g_{10} (0.36) = \frac{1}{3} lo g_{10} (\frac{9}{25})

= \frac{1}{4} lo g_{10} (\frac{97}{200}) + \frac{1}{3} lo g_{10} (\frac{9}{25})

\frac{1}{4} lo g_{10} (\frac{97}{200}) = lo g_{10} ((\frac{97}{200})^{\frac{1}{4}})

\frac{1}{3} lo g_{10} (\frac{9}{25}) = lo g_{10} ((\frac{9}{25})^{\frac{1}{3}})

= lo g_{10} ((\frac{97}{200})^{\frac{1}{4}}) + lo g_{10} ((\frac{9}{25})^{\frac{1}{3}})

使用对数计算法则:

lo g_{a} (x) + lo g_{a} (y) = lo g_{a} (x y)

lo g_{10} ((\frac{97}{200})^{\frac{1}{4}}) + lo g_{10} ((\frac{9}{25})^{\frac{1}{3}}) = lo g_{10} ((\frac{97}{200})^{\frac{1}{4}} (\frac{9}{25})^{\frac{1}{3}})

= lo g_{10} ((\frac{97}{200})^{\frac{1}{4}} (\frac{9}{25})^{\frac{1}{3}})

(\frac{97}{200})^{\frac{1}{4}} (\frac{9}{25})^{\frac{1}{3}} = 4 \frac{97}{200} 3 \frac{9}{25}

= lo g_{10} (4 \frac{97}{200} 3 \frac{9}{25})

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