簡約 (2ln(3)-ln(2))/(2ln(3)-3ln(2))

解

簡約

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

\frac{ln ( \frac{9}{2} )}{ln ( \frac{9}{8} )}

十進法表記

12.76989 \dots

解答ステップ

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

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 ( 3 ) - 3 ln ( 2 )}

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

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

3^{2} = 9

= \frac{ln ( \frac{9}{2} )}{2 ln ( 3 ) - 3 ln ( 2 )}

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 ( \frac{9}{2} )}{ln ( 3 ^{2} ) - 3 ln ( 2 )}

対数の規則を適用する:

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

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

= \frac{ln ( \frac{9}{2} )}{ln ( 3 ^{2} ) - ln ( 2 ^{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 (3^{2}) - ln (2^{3}) = ln (\frac{3 ^{2}}{2 ^{3}})

= \frac{ln ( \frac{9}{2} )}{ln ( \frac{3 ^{2}}{2 ^{3}} )}

\frac{3 ^{2}}{2 ^{3}} = \frac{9}{8}

= \frac{ln ( \frac{9}{2} )}{ln ( \frac{9}{8} )}