簡約 ln((pi/447^2)/(pi/430^2))

解

簡約

ln (\frac{\frac{π}{4} \cdot 4 7 ^{2}}{\frac{π}{4} \cdot 3 0 ^{2}})

ln (π) - 2 ln (2) + 2 ln (47) - ln (\frac{π}{4}) - 2 ln (30)

十進法表記

0.89790 \dots

解答ステップ

ln (\frac{\frac{π}{4} \cdot 4 7 ^{2}}{\frac{π}{4} \cdot 3 0 ^{2}})

対数の規則を適用する:

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

ln (\frac{\frac{π}{4} \cdot 4 7 ^{2}}{\frac{π}{4} \cdot 3 0 ^{2}}) = ln (\frac{π}{4} \cdot 4 7^{2}) - ln (\frac{π}{4} \cdot 3 0^{2})

= ln (4 7^{2} \cdot \frac{π}{4}) - ln (3 0^{2} \cdot \frac{π}{4})

対数の規則を適用する:

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

ln (\frac{π}{4} \cdot 4 7^{2}) = ln (\frac{π}{4}) + ln (4 7^{2}), ln (\frac{π}{4} \cdot 3 0^{2}) = ln (\frac{π}{4}) + ln (3 0^{2})

= ln (\frac{π}{4}) + ln (4 7^{2}) - (ln (3 0^{2}) + ln (\frac{π}{4}))

対数の規則を適用する:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (4 7^{2}) = 2 ln (47)

= ln (\frac{π}{4}) + 2 ln (47) - (ln (\frac{π}{4}) + ln (3 0^{2}))

対数の規則を適用する:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (3 0^{2}) = 2 ln (30)

= ln (\frac{π}{4}) + 2 ln (47) - (ln (\frac{π}{4}) + 2 ln (30))

対数の規則を適用する:

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

ln (\frac{π}{4}) = ln (π) - ln (4)

= ln (π) - ln (4) + 2 ln (47) - (2 ln (30) + ln (\frac{π}{4}))

ln (4) = 2 ln (2)

= ln (π) - 2 ln (2) + 2 ln (47) - (2 ln (30) + ln (\frac{π}{4}))

- (ln (\frac{π}{4}) + 2 ln (30)) : - ln (\frac{π}{4}) - 2 ln (30)

= ln (π) - 2 ln (2) + 2 ln (47) - ln (\frac{π}{4}) - 2 ln (30)