簡約 1/3 log_{10}(125)-1/2 log_{10}(64)

解

簡約

\frac{1}{3} lo g_{10} (125) - \frac{1}{2} lo g_{10} (64)

lo g_{10} (\frac{5}{8})

十進法表記

- 0.20411 \dots

解答ステップ

\frac{1}{3} lo g_{10} (125) - \frac{1}{2} lo g_{10} (64)

\frac{1}{3} lo g_{10} (125) = lo g_{10} (12 5^{\frac{1}{3}})

- \frac{1}{2} lo g_{10} (64) = - lo g_{10} (6 4^{\frac{1}{2}})

= lo g_{10} (12 5^{\frac{1}{3}}) - lo g_{10} (6 4^{\frac{1}{2}})

対数の規則を適用する:

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

lo g_{10} (12 5^{\frac{1}{3}}) - lo g_{10} (6 4^{\frac{1}{2}}) = lo g_{10} (\frac{12 5 ^{\frac{1}{3}}}{6 4 ^{\frac{1}{2}}})

= lo g_{10} (\frac{12 5 ^{\frac{1}{3}}}{6 4 ^{\frac{1}{2}}})

\frac{12 5 ^{\frac{1}{3}}}{6 4 ^{\frac{1}{2}}} = \frac{5}{8}

= lo g_{10} (\frac{5}{8})