簡約 ((-4ln(5)+2ln(3)))/(7ln(5)-ln(3))

解

簡約

\frac{( - 4 ln ( 5 ) + 2 ln ( 3 ) )}{7 ln ( 5 ) - ln ( 3 )}

\frac{ln ( \frac{9}{625} )}{ln ( \frac{78125}{3} )}

十進法表記

- 0.41706 \dots

解答ステップ

\frac{( - 4 ln ( 5 ) + 2 ln ( 3 ) )}{7 ln ( 5 ) - ln ( 3 )}

対数の規則を適用する:

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

4 ln (5) = ln (5^{4})

= \frac{( - ln ( 5 ^{4} ) + 2 ln ( 3 ) )}{7 ln ( 5 ) - 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 ( 5 ^{4} ) + ln ( 3 ^{2} ) )}{7 ln ( 5 ) - 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 (3^{2}) - ln (5^{4}) = ln (\frac{3 ^{2}}{5 ^{4}})

= \frac{ln ( \frac{3 ^{2}}{5 ^{4}} )}{7 ln ( 5 ) - ln ( 3 )}

\frac{3 ^{2}}{5 ^{4}} = \frac{9}{625}

= \frac{ln ( \frac{9}{625} )}{7 ln ( 5 ) - ln ( 3 )}

対数の規則を適用する:

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

7 ln (5) = ln (5^{7})

= \frac{ln ( \frac{9}{625} )}{ln ( 5 ^{7} ) - 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 (5^{7}) - ln (3) = ln (\frac{5 ^{7}}{3})

= \frac{ln ( \frac{9}{625} )}{ln ( \frac{5 ^{7}}{3} )}

5^{7} = 78125

= \frac{ln ( \frac{9}{625} )}{ln ( \frac{78125}{3} )}