簡約 (ln(5)-2ln(2))/(ln(3)-ln(2))

解

簡約

\frac{ln ( 5 ) - 2 ln ( 2 )}{ln ( 3 ) - ln ( 2 )}

\frac{ln ( \frac{5}{4} )}{ln ( \frac{3}{2} )}

十進法表記

0.55033 \dots

解答ステップ

\frac{ln ( 5 ) - 2 ln ( 2 )}{ln ( 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 (2) = ln (2^{2})

= \frac{ln ( 5 ) - ln ( 2 ^{2} )}{ln ( 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 (5) - ln (2^{2}) = ln (\frac{5}{2 ^{2}})

= \frac{ln ( \frac{5}{2 ^{2}} )}{ln ( 3 ) - ln ( 2 )}

2^{2} = 4

= \frac{ln ( \frac{5}{4} )}{ln ( 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) - ln (2) = ln (\frac{3}{2})

= \frac{ln ( \frac{5}{4} )}{ln ( \frac{3}{2} )}