What is simplify-0.45log_{2}(0.45)-0.55log_{2}(0.55) ?

The solution to simplify-0.45log_{2}(0.45)-0.55log_{2}(0.55) is -log_{2}((11^{0.55}*3^{0.9})/5)+2

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simplify-0.45log_{2}(0.45)-0.55log_{2}(0.55)

simplify

- 0.45 lo g_{2} (0.45) - 0.55 lo g_{2} (0.55)

- lo g_{2} (\frac{1 1 ^{0.55} \cdot 3 ^{0.9}}{5}) + 2

Decimal Notation

0.99277 \dots

Solution steps

- 0.45 lo g_{2} (0.45) - 0.55 lo g_{2} (0.55)

- 0.45 lo g_{2} (0.45) = - 0.45 lo g_{2} (\frac{9}{20})

lo g_{2} (0.55) = lo g_{2} (\frac{11}{20})

= - 0.45 lo g_{2} (\frac{9}{20}) - 0.55 lo g_{2} (\frac{11}{20})

- 0.45 lo g_{2} (\frac{9}{20}) = - lo g_{2} ((\frac{9}{20})^{0.45})

- 0.55 lo g_{2} (\frac{11}{20}) = - lo g_{2} ((\frac{11}{20})^{0.55})

= - lo g_{2} ((\frac{9}{20})^{0.45}) - lo g_{2} ((\frac{11}{20})^{0.55})

Apply log rule:

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

- lo g_{2} ((\frac{11}{20})^{0.55}) - lo g_{2} ((\frac{9}{20})^{0.45}) = lo g_{2} ((\frac{11}{20})^{0.55} (\frac{9}{20})^{0.45})

= - lo g_{2} ((\frac{11}{20})^{0.55} (\frac{9}{20})^{0.45})

Simplify

(\frac{11}{20})^{0.55} (\frac{9}{20})^{0.45} : \frac{1 1 ^{0.55} \cdot 9 ^{0.45}}{20}

= - lo g_{2} (\frac{1 1 ^{0.55} \cdot 9 ^{0.45}}{20})

Apply log rule:

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

lo g_{2} (\frac{1 1 ^{0.55} \cdot 9 ^{0.45}}{20}) = lo g_{2} (1 1^{0.55} \cdot 9^{0.45}) - lo g_{2} (20)

= - (lo g_{2} (1 1^{0.55} \cdot 9^{0.45}) - lo g_{2} (20))

lo g_{2} (20) = 2 + lo g_{2} (5)

= - (lo g_{2} (1 1^{0.55} \cdot 9^{0.45}) - (2 + lo g_{2} (5)))

Apply the distributive law:

- (a + b) = - a - b

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

= - (lo g_{2} (1 1^{0.55} \cdot 9^{0.45}) - 2 - lo g_{2} (5))

lo g_{2} (1 1^{0.55} \cdot 9^{0.45}) = 0.55 lo g_{2} (11) + 0.9 lo g_{2} (3)

= - ((0.55 lo g_{2} (11) + 0.9 lo g_{2} (3)) - 2 - lo g_{2} (5))

Apply rule:

(a) = a

(0.55 lo g_{2} (11) + 0.9 lo g_{2} (3)) = 0.55 lo g_{2} (11) + 0.9 lo g_{2} (3)

= - (0.55 lo g_{2} (11) + 0.9 lo g_{2} (3) - 2 - lo g_{2} (5))

0.55 lo g_{2} (11) = lo g_{2} (1 1^{0.55})

0.9 lo g_{2} (3) = lo g_{2} (3^{0.9})

= - (lo g_{2} (1 1^{0.55}) + lo g_{2} (3^{0.9}) - 2 - lo g_{2} (5))

lo g_{2} (1 1^{0.55}) + lo g_{2} (3^{0.9}) - lo g_{2} (5) = lo g_{2} (\frac{1 1 ^{0.55} \cdot 3 ^{0.9}}{5})

= - (lo g_{2} (\frac{1 1 ^{0.55} \cdot 3 ^{0.9}}{5}) - 2)

Apply the distributive law:

- (a - b) = - a + b

- (lo g_{2} (\frac{1 1 ^{0.55} \cdot 3 ^{0.9}}{5}) - 2) = - lo g_{2} (\frac{1 1 ^{0.55} \cdot 3 ^{0.9}}{5}) + 2

= - lo g_{2} (\frac{1 1 ^{0.55} \cdot 3 ^{0.9}}{5}) + 2

lo g_{4} (2 x^{2})

lo g_{10} (\frac{5}{x + 1})

ln (z + r^{2} + z^{2}) - ln (- z + r^{2} + z^{2})

lo g_{a} (\frac{x ^{6}}{y z ^{7}})

lo g_{10} (3 x + 5) - lo g_{10} (x)

What is simplify-0.45log_{2}(0.45)-0.55log_{2}(0.55) ?
The solution to simplify-0.45log_{2}(0.45)-0.55log_{2}(0.55) is -log_{2}((11^{0.55}*3^{0.9})/5)+2