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Beliebt Algebra >

vereinfachen-0.43log_{2}(0.43)-0.57log_{2}(0.57)

Lösung

vereinfachen

- 0.43 lo g_{2} (0.43) - 0.57 lo g_{2} (0.57)

Lösung

- lo g_{2} (\frac{5 7 ^{0.57} \cdot 4 3 ^{0.43}}{25}) + 2

+1

Dezimale

0.98581 \dots

Schritte zur Lösung

- 0.43 lo g_{2} (0.43) - 0.57 lo g_{2} (0.57)

- 0.43 lo g_{2} (0.43) = - 0.43 lo g_{2} (\frac{43}{100})

0.57 = \frac{57}{100}

= - 0.43 lo g_{2} (\frac{43}{100}) - 0.57 lo g_{2} (\frac{57}{100})

- 0.43 lo g_{2} (\frac{43}{100}) = - lo g_{2} ((\frac{43}{100})^{0.43})

- 0.57 lo g_{2} (\frac{57}{100}) = - lo g_{2} ((\frac{57}{100})^{0.57})

= - lo g_{2} ((\frac{43}{100})^{0.43}) - lo g_{2} ((\frac{57}{100})^{0.57})

Wende die log Regel an:

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

- lo g_{2} ((\frac{57}{100})^{0.57}) - lo g_{2} ((\frac{43}{100})^{0.43}) = lo g_{2} ((\frac{57}{100})^{0.57} (\frac{43}{100})^{0.43})

= - lo g_{2} ((\frac{57}{100})^{0.57} (\frac{43}{100})^{0.43})

Vereinfache

(\frac{57}{100})^{0.57} (\frac{43}{100})^{0.43} : \frac{5 7 ^{0.57} \cdot 4 3 ^{0.43}}{100}

= - lo g_{2} (\frac{5 7 ^{0.57} \cdot 4 3 ^{0.43}}{100})

Wende die log Regel an:

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

lo g_{2} (\frac{5 7 ^{0.57} \cdot 4 3 ^{0.43}}{100}) = lo g_{2} (5 7^{0.57} \cdot 4 3^{0.43}) - lo g_{2} (100)

= - (lo g_{2} (5 7^{0.57} \cdot 4 3^{0.43}) - lo g_{2} (100))

lo g_{2} (5 7^{0.57} \cdot 4 3^{0.43}) = 0.57 lo g_{2} (57) + 0.43 lo g_{2} (43)

= - ((0.57 lo g_{2} (57) + 0.43 lo g_{2} (43)) - lo g_{2} (100))

Apply rule:

(a) = a

(0.57 lo g_{2} (57) + 0.43 lo g_{2} (43)) = 0.57 lo g_{2} (57) + 0.43 lo g_{2} (43)

= - (0.57 lo g_{2} (57) + 0.43 lo g_{2} (43) - lo g_{2} (100))

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

= - (0.57 lo g_{2} (57) + 0.43 lo g_{2} (43) - (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)

= - (0.57 lo g_{2} (57) + 0.43 lo g_{2} (43) - 2 - 2 lo g_{2} (5))

0.57 lo g_{2} (57) = lo g_{2} (5 7^{0.57})

0.43 lo g_{2} (43) = lo g_{2} (4 3^{0.43})

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

= - (lo g_{2} (5 7^{0.57}) + lo g_{2} (4 3^{0.43}) - 2 - lo g_{2} (5^{2}))

lo g_{2} (5 7^{0.57}) + lo g_{2} (4 3^{0.43}) - lo g_{2} (5^{2}) = lo g_{2} (\frac{5 7 ^{0.57} \cdot 4 3 ^{0.43}}{25})

= - (lo g_{2} (\frac{5 7 ^{0.57} \cdot 4 3 ^{0.43}}{25}) - 2)

Wende das Distributivgesetz an:

- (a - b) = - a + b

- (lo g_{2} (\frac{5 7 ^{0.57} \cdot 4 3 ^{0.43}}{25}) - 2) = - lo g_{2} (\frac{5 7 ^{0.57} \cdot 4 3 ^{0.43}}{25}) + 2

= - lo g_{2} (\frac{5 7 ^{0.57} \cdot 4 3 ^{0.43}}{25}) + 2

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