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Beliebt Algebra >

vereinfachen-0.67log_{2}(0.67)-0.33log_{2}(0.33)

Lösung

vereinfachen

- 0.67 lo g_{2} (0.67) - 0.33 lo g_{2} (0.33)

Lösung

- lo g_{2} (\frac{3 3 ^{0.33} \cdot 6 7 ^{0.67}}{25}) + 2

+1

Dezimale

0.91492 \dots

Schritte zur Lösung

- 0.67 lo g_{2} (0.67) - 0.33 lo g_{2} (0.33)

- 0.67 lo g_{2} (0.67) = - 0.67 lo g_{2} (\frac{67}{100})

0.33 = \frac{33}{100}

= - 0.67 lo g_{2} (\frac{67}{100}) - 0.33 lo g_{2} (\frac{33}{100})

- 0.67 lo g_{2} (\frac{67}{100}) = - lo g_{2} ((\frac{67}{100})^{0.67})

- 0.33 lo g_{2} (\frac{33}{100}) = - lo g_{2} ((\frac{33}{100})^{0.33})

= - lo g_{2} ((\frac{67}{100})^{0.67}) - lo g_{2} ((\frac{33}{100})^{0.33})

Wende die log Regel an:

lo g_{a} (x) + lo g_{a} (y) = lo g_{a} (x y)

- lo g_{2} ((\frac{33}{100})^{0.33}) - lo g_{2} ((\frac{67}{100})^{0.67}) = lo g_{2} ((\frac{33}{100})^{0.33} (\frac{67}{100})^{0.67})

= - lo g_{2} ((\frac{33}{100})^{0.33} (\frac{67}{100})^{0.67})

Vereinfache

(\frac{33}{100})^{0.33} (\frac{67}{100})^{0.67} : \frac{3 3 ^{0.33} \cdot 6 7 ^{0.67}}{100}

= - lo g_{2} (\frac{3 3 ^{0.33} \cdot 6 7 ^{0.67}}{100})

Wende die log Regel an:

lo g_{a} (\frac{x}{y}) = lo g_{a} (x) - lo g_{a} (y)

lo g_{2} (\frac{3 3 ^{0.33} \cdot 6 7 ^{0.67}}{100}) = lo g_{2} (3 3^{0.33} \cdot 6 7^{0.67}) - lo g_{2} (100)

= - (lo g_{2} (3 3^{0.33} \cdot 6 7^{0.67}) - lo g_{2} (100))

lo g_{2} (3 3^{0.33} \cdot 6 7^{0.67}) = 0.33 lo g_{2} (33) + 0.67 lo g_{2} (67)

= - ((0.33 lo g_{2} (33) + 0.67 lo g_{2} (67)) - lo g_{2} (100))

Apply rule:

(a) = a

(0.33 lo g_{2} (33) + 0.67 lo g_{2} (67)) = 0.33 lo g_{2} (33) + 0.67 lo g_{2} (67)

= - (0.33 lo g_{2} (33) + 0.67 lo g_{2} (67) - lo g_{2} (100))

lo g_{2} (100) = 2 + 2 lo g_{2} (5)

= - (0.33 lo g_{2} (33) + 0.67 lo g_{2} (67) - (2 + 2 lo g_{2} (5)))

Wende das Distributivgesetz an:

- (a + b) = - a - b

- (2 + 2 lo g_{2} (5)) = - 2 - 2 lo g_{2} (5)

= - (0.33 lo g_{2} (33) + 0.67 lo g_{2} (67) - 2 - 2 lo g_{2} (5))

0.33 lo g_{2} (33) = lo g_{2} (3 3^{0.33})

0.67 lo g_{2} (67) = lo g_{2} (6 7^{0.67})

- 2 lo g_{2} (5) = - lo g_{2} (5^{2})

= - (lo g_{2} (3 3^{0.33}) + lo g_{2} (6 7^{0.67}) - 2 - lo g_{2} (5^{2}))

lo g_{2} (3 3^{0.33}) + lo g_{2} (6 7^{0.67}) - lo g_{2} (5^{2}) = lo g_{2} (\frac{3 3 ^{0.33} \cdot 6 7 ^{0.67}}{25})

= - (lo g_{2} (\frac{3 3 ^{0.33} \cdot 6 7 ^{0.67}}{25}) - 2)

Wende das Distributivgesetz an:

- (a - b) = - a + b

- (lo g_{2} (\frac{3 3 ^{0.33} \cdot 6 7 ^{0.67}}{25}) - 2) = - lo g_{2} (\frac{3 3 ^{0.33} \cdot 6 7 ^{0.67}}{25}) + 2

= - lo g_{2} (\frac{3 3 ^{0.33} \cdot 6 7 ^{0.67}}{25}) + 2

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