簡約 ln(3)(-1-1/2+1/8+3/4)+ln(2)(1-1/4-3/4)

解

簡約

ln (3) (- 1 - \frac{1}{2} + \frac{1}{8} + \frac{3}{4}) + ln (2) (1 - \frac{1}{4} - \frac{3}{4})

ln (\frac{3 ^{\frac{3}{8}}}{3})

十進法表記

- 0.68663 \dots

解答ステップ

ln (3) (- 1 - \frac{1}{2} + \frac{1}{8} + \frac{3}{4}) + ln (2) (1 - \frac{1}{4} - \frac{3}{4})

対数の規則を適用する:

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

ln (3) (\frac{1}{8} + \frac{3}{4} - 1 - \frac{1}{2}) = ln (3^{(\frac{1}{8} + \frac{3}{4} - 1 - \frac{1}{2})})

= ln (3^{(\frac{1}{8} + \frac{3}{4} - 1 - \frac{1}{2})}) + ln (2) (1 - \frac{1}{4} - \frac{3}{4})

対数の規則を適用する:

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

ln (2) (1 - \frac{1}{4} - \frac{3}{4}) = ln (2^{(1 - \frac{1}{4} - \frac{3}{4})})

= ln (3^{(\frac{1}{8} + \frac{3}{4} - 1 - \frac{1}{2})}) + ln (2^{(1 - \frac{1}{4} - \frac{3}{4})})

対数の規則を適用する:

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

ln (3^{(- 1 - \frac{1}{2} + \frac{1}{8} + \frac{3}{4})}) + ln (2^{(1 - \frac{1}{4} - \frac{3}{4})}) = ln (3^{\frac{1}{8} + \frac{3}{4} - 1 - \frac{1}{2}} \cdot 2^{1 - \frac{1}{4} - \frac{3}{4}})

= ln (3^{\frac{1}{8} + \frac{3}{4} - 1 - \frac{1}{2}} \cdot 2^{1 - \frac{1}{4} - \frac{3}{4}})

3^{(- 1 - \frac{1}{2} + \frac{1}{8} + \frac{3}{4})} \cdot 2^{(1 - \frac{1}{4} - \frac{3}{4})} = \frac{3 ^{\frac{3}{8}}}{3}

= ln (\frac{3 ^{\frac{3}{8}}}{3})