簡約-(24ln(2)-3ln(15))/(2(ln(15)-9ln(2)))

解

簡約

- \frac{24 ln ( 2 ) - 3 ln ( 15 )}{2 ( ln ( 15 ) - 9 ln ( 2 ) )}

- \frac{ln ( \frac{16777216}{3375} )}{2 ln ( \frac{15}{512} )}

十進法表記

1.20548 \dots

解答ステップ

- \frac{24 ln ( 2 ) - 3 ln ( 15 )}{2 ( ln ( 15 ) - 9 ln ( 2 ) )}

対数の規則を適用する:

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

24 ln (2) = ln (2^{24})

= - \frac{ln ( 2 ^{24} ) - 3 ln ( 15 )}{2 ( ln ( 15 ) - 9 ln ( 2 ) )}

対数の規則を適用する:

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

3 ln (15) = ln (1 5^{3})

= - \frac{ln ( 2 ^{24} ) - ln ( 1 5 ^{3} )}{2 ( ln ( 15 ) - 9 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 (2^{24}) - ln (1 5^{3}) = ln (\frac{2 ^{24}}{1 5 ^{3}})

= - \frac{ln ( \frac{2 ^{24}}{1 5 ^{3}} )}{2 ( ln ( 15 ) - 9 ln ( 2 ) )}

\frac{2 ^{24}}{1 5 ^{3}} = \frac{16777216}{3375}

= - \frac{ln ( \frac{16777216}{3375} )}{2 ( ln ( 15 ) - 9 ln ( 2 ) )}

対数の規則を適用する:

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

9 ln (2) = ln (2^{9})

= - \frac{ln ( \frac{16777216}{3375} )}{2 ( ln ( 15 ) - ln ( 2 ^{9} ) )}

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 (15) - ln (2^{9}) = ln (\frac{15}{2 ^{9}})

= - \frac{ln ( \frac{16777216}{3375} )}{2 ln ( \frac{15}{2 ^{9}} )}

2^{9} = 512

= - \frac{ln ( \frac{16777216}{3375} )}{2 ln ( \frac{15}{512} )}