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Beliebt Algebra >

vereinfachen 5/6 log_{2}(5/6)+1/6 log_{2}(1/6)

Lösung

vereinfachen

\frac{5}{6} lo g_{2} (\frac{5}{6}) + \frac{1}{6} lo g_{2} (\frac{1}{6})

Lösung

\frac{1}{6} lo g_{2} (\frac{3125}{46656})

+1

Dezimale

- 0.65002 \dots

Schritte zur Lösung

\frac{5}{6} lo g_{2} (\frac{5}{6}) + \frac{1}{6} lo g_{2} (\frac{1}{6})

Wende die log Regel an:

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

\frac{5}{6} lo g_{2} (\frac{5}{6}) = lo g_{2} ((\frac{5}{6})^{\frac{5}{6}})

= lo g_{2} ((\frac{5}{6})^{\frac{5}{6}}) + \frac{1}{6} lo g_{2} (\frac{1}{6})

Wende die log Regel an:

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

\frac{1}{6} lo g_{2} (\frac{1}{6}) = lo g_{2} ((\frac{1}{6})^{\frac{1}{6}})

= lo g_{2} ((\frac{5}{6})^{\frac{5}{6}}) + lo g_{2} ((\frac{1}{6})^{\frac{1}{6}})

Wende die log Regel an:

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

lo g_{2} ((\frac{5}{6})^{\frac{5}{6}}) + lo g_{2} ((\frac{1}{6})^{\frac{1}{6}}) = lo g_{2} ((\frac{5}{6})^{\frac{5}{6}} (\frac{1}{6})^{\frac{1}{6}})

= lo g_{2} ((\frac{5}{6})^{\frac{5}{6}} (\frac{1}{6})^{\frac{1}{6}})

Wende Exponentenregel an:

a^{m} b^{m} = (ab)^{m}

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

= lo g_{2} 6 (\frac{5}{6})^{5} \frac{1}{6}

Schreibe um

= lo g_{2} ((\frac{5}{6})^{5} \frac{1}{6})^{\frac{1}{6}}

Wende log Regel an

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

angenommen

x \geq 0

= \frac{1}{6} lo g_{2} ((\frac{5}{6})^{5} \frac{1}{6})

(\frac{5}{6})^{5} = \frac{5 ^{5}}{6 ^{5}}

= \frac{1}{6} lo g_{2} (\frac{1}{6} \cdot \frac{5 ^{5}}{6 ^{5}})

Multipliziere

\frac{5 ^{5}}{6 ^{5}} \cdot \frac{1}{6} : \frac{3125}{46656}

= \frac{1}{6} lo g_{2} (\frac{3125}{46656})

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