化简 ln(pi/4)*sqrt(pi/4)

解答

化简

ln (\frac{π}{4}) \cdot \frac{π}{4}

\frac{π}{2} ln (π) - π ln (2)

十进制

- 0.21408 \dots

求解步骤

ln (\frac{π}{4}) \frac{π}{4}

使用对数计算法则:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

ln (\frac{π}{4}) = ln (π) - ln (4)

= \frac{π}{4} (ln (π) - ln (4))

\frac{π}{4} = \frac{π}{2}

= \frac{π}{2} (ln (π) - ln (4))

ln (4) = 2 ln (2)

= \frac{π}{2} (ln (π) - 2 ln (2))

= \frac{π}{2} (ln (π) - 2 ln (2))

使用分配律:

a (b - c) = ab - a c

a = \frac{π}{2}, b = ln (π), c = 2 ln (2)

= \frac{π}{2} ln (π) - \frac{π}{2} \cdot 2 ln (2)

= \frac{π}{2} ln (π) - 2 \cdot \frac{π}{2} ln (2)

2 \cdot \frac{π}{2} ln (2) = π ln (2)

= \frac{π}{2} ln (π) - π ln (2)

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