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Beliebt Algebra >

vereinfachen-1/16 log_{e}(1/16)-15/16 log_{e}(15/16)

Lösung

vereinfachen

- \frac{1}{16} lo g_{e} (\frac{1}{16}) - \frac{15}{16} lo g_{e} (\frac{15}{16})

Lösung

- \frac{1}{16} ln (\frac{1 5 ^{15}}{1 6 ^{16}})

+1

Dezimale

0.23379 \dots

Schritte zur Lösung

- \frac{1}{16} lo g_{e} (\frac{1}{16}) - \frac{15}{16} lo g_{e} (\frac{15}{16})

Wende die log Regel an:

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

\frac{1}{16} lo g_{e} (\frac{1}{16}) = lo g_{e} ((\frac{1}{16})^{\frac{1}{16}})

= - lo g_{e} ((\frac{1}{16})^{\frac{1}{16}}) - \frac{15}{16} lo g_{e} (\frac{15}{16})

Wende die log Regel an:

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

\frac{15}{16} lo g_{e} (\frac{15}{16}) = lo g_{e} ((\frac{15}{16})^{\frac{15}{16}})

= - lo g_{e} ((\frac{1}{16})^{\frac{1}{16}}) - lo g_{e} ((\frac{15}{16})^{\frac{15}{16}})

Schreibe

- lo g_{e} ((\frac{1}{16})^{\frac{1}{16}}) - lo g_{e} ((\frac{15}{16})^{\frac{15}{16}})

um:

- (lo g_{e} ((\frac{1}{16})^{\frac{1}{16}}) + lo g_{e} ((\frac{15}{16})^{\frac{15}{16}}))

= - (lo g_{e} ((\frac{1}{16})^{\frac{1}{16}}) + lo g_{e} ((\frac{15}{16})^{\frac{15}{16}}))

Wende die log Regel an:

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

lo g_{e} ((\frac{1}{16})^{\frac{1}{16}}) + lo g_{e} ((\frac{15}{16})^{\frac{15}{16}}) = lo g_{e} ((\frac{1}{16})^{\frac{1}{16}} (\frac{15}{16})^{\frac{15}{16}})

= - lo g_{e} ((\frac{1}{16})^{\frac{1}{16}} (\frac{15}{16})^{\frac{15}{16}})

Wende die log Regel an:

lo g_{e} (x) = ln (x)

= - ln ((\frac{1}{16})^{\frac{1}{16}} (\frac{15}{16})^{\frac{15}{16}})

Vereinfache

ln ((\frac{1}{16})^{\frac{1}{16}} (\frac{15}{16})^{\frac{15}{16}}) : \frac{1}{16} ln (\frac{1}{16} (\frac{15}{16})^{15})

= - \frac{1}{16} ln (\frac{1}{16} (\frac{15}{16})^{15})

(\frac{15}{16})^{15} = \frac{1 5 ^{15}}{1 6 ^{15}}

= - \frac{1}{16} ln (\frac{1}{16} \cdot \frac{1 5 ^{15}}{1 6 ^{15}})

Multipliziere

\frac{1}{16} \cdot \frac{1 5 ^{15}}{1 6 ^{15}} : \frac{1 5 ^{15}}{1 6 ^{16}}

= - \frac{1}{16} ln (\frac{1 5 ^{15}}{1 6 ^{16}})

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