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Beliebt Algebra >

vereinfachen-1/10 log_{2}(1/10)-9/10 log_{2}(9/10)

Lösung

vereinfachen

- \frac{1}{10} lo g_{2} (\frac{1}{10}) - \frac{9}{10} lo g_{2} (\frac{9}{10})

Lösung

- \frac{1}{10} lo g_{2} (\frac{387420489}{1 0 ^{10}})

+1

Dezimale

0.46899 \dots

Schritte zur Lösung

- \frac{1}{10} lo g_{2} (\frac{1}{10}) - \frac{9}{10} lo g_{2} (\frac{9}{10})

Wende die log Regel an:

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

\frac{1}{10} lo g_{2} (\frac{1}{10}) = lo g_{2} ((\frac{1}{10})^{\frac{1}{10}})

= - lo g_{2} ((\frac{1}{10})^{\frac{1}{10}}) - \frac{9}{10} lo g_{2} (\frac{9}{10})

Wende die log Regel an:

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

\frac{9}{10} lo g_{2} (\frac{9}{10}) = lo g_{2} ((\frac{9}{10})^{\frac{9}{10}})

= - lo g_{2} ((\frac{1}{10})^{\frac{1}{10}}) - lo g_{2} ((\frac{9}{10})^{\frac{9}{10}})

Schreibe

- lo g_{2} ((\frac{1}{10})^{\frac{1}{10}}) - lo g_{2} ((\frac{9}{10})^{\frac{9}{10}})

um:

- (lo g_{2} ((\frac{1}{10})^{\frac{1}{10}}) + lo g_{2} ((\frac{9}{10})^{\frac{9}{10}}))

= - (lo g_{2} ((\frac{1}{10})^{\frac{1}{10}}) + lo g_{2} ((\frac{9}{10})^{\frac{9}{10}}))

Wende die log Regel an:

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

lo g_{2} ((\frac{1}{10})^{\frac{1}{10}}) + lo g_{2} ((\frac{9}{10})^{\frac{9}{10}}) = lo g_{2} ((\frac{1}{10})^{\frac{1}{10}} (\frac{9}{10})^{\frac{9}{10}})

= - lo g_{2} ((\frac{1}{10})^{\frac{1}{10}} (\frac{9}{10})^{\frac{9}{10}})

Vereinfache

lo g_{2} ((\frac{1}{10})^{\frac{1}{10}} (\frac{9}{10})^{\frac{9}{10}}) : \frac{1}{10} lo g_{2} ((\frac{9}{10})^{9} \frac{1}{10})

= - \frac{1}{10} lo g_{2} ((\frac{9}{10})^{9} \frac{1}{10})

(\frac{9}{10})^{9} = \frac{9 ^{9}}{1 0 ^{9}}

= - \frac{1}{10} lo g_{2} (\frac{1}{10} \cdot \frac{9 ^{9}}{1 0 ^{9}})

Multipliziere

\frac{1}{10} \cdot \frac{9 ^{9}}{1 0 ^{9}} : \frac{387420489}{1 0 ^{10}}

= - \frac{1}{10} lo g_{2} (\frac{387420489}{1 0 ^{10}})

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