簡約 (2ln(3)+ln(2))/(2ln(2)-ln(3))

解

簡約

\frac{2 ln ( 3 ) + ln ( 2 )}{2 ln ( 2 ) - ln ( 3 )}

\frac{ln ( 18 )}{ln ( \frac{4}{3} )}

十進法表記

10.04710 \dots

解答ステップ

\frac{2 ln ( 3 ) + ln ( 2 )}{2 ln ( 2 ) - ln ( 3 )}

対数の規則を適用する:

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

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

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

対数の規則を適用する:

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

ln (3^{2}) + ln (2) = ln (3^{2} \cdot 2)

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

3^{2} \cdot 2 = 18

= \frac{ln ( 18 )}{2 ln ( 2 ) - ln ( 3 )}

対数の規則を適用する:

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

2 ln (2) = ln (2^{2})

= \frac{ln ( 18 )}{ln ( 2 ^{2} ) - ln ( 3 )}

対数の規則を適用する:

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

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

= \frac{ln ( 18 )}{ln ( \frac{2 ^{2}}{3} )}

2^{2} = 4

= \frac{ln ( 18 )}{ln ( \frac{4}{3} )}