簡約 0.4log_{10}(1/(0.4))+(1-0.4)log_{10}(1/(1-0.4))

解

簡約

0.4 \cdot lo g_{10} (\frac{1}{0.4}) + (1 - 0.4) \cdot lo g_{10} (\frac{1}{1 - 0.4})

lo g_{10} (1.96013 \dots)

十進法表記

0.29228 \dots

解答ステップ

0.4 lo g_{10} (\frac{1}{0.4}) + (1 - 0.4) lo g_{10} (\frac{1}{1 - 0.4})

対数の規則を適用する:

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

0.4 lo g_{10} (\frac{1}{0.4}) = lo g_{10} ((\frac{1}{0.4})^{0.4})

= lo g_{10} ((\frac{1}{0.4})^{0.4}) + lo g_{10} (\frac{1}{1 - 0.4}) (1 - 0.4)

対数の規則を適用する:

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

lo g_{10} (\frac{1}{1 - 0.4}) (1 - 0.4) = lo g_{10} ((\frac{1}{1 - 0.4})^{(1 - 0.4)})

= lo g_{10} ((\frac{1}{0.4})^{0.4}) + lo g_{10} ((\frac{1}{1 - 0.4})^{(1 - 0.4)})

対数の規則を適用する:

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

lo g_{10} ((\frac{1}{0.4})^{0.4}) + lo g_{10} ((\frac{1}{1 - 0.4})^{(1 - 0.4)}) = lo g_{10} ((\frac{1}{0.4})^{0.4} (\frac{1}{1 - 0.4})^{1 - 0.4})

= lo g_{10} ((\frac{1}{0.4})^{0.4} (\frac{1}{1 - 0.4})^{1 - 0.4})

(\frac{1}{0.4})^{0.4} (\frac{1}{1 - 0.4})^{(1 - 0.4)} = 1.96013 \dots

= lo g_{10} (1.96013 \dots)