What is simplify-0.3log_{2}(0.3)-0.7log_{2}(0.7) ?

The solution to simplify-0.3log_{2}(0.3)-0.7log_{2}(0.7) is -log_{2}((7^{0.7}*3^{0.3})/5)+1

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simplify-0.3log_{2}(0.3)-0.7log_{2}(0.7)

simplify

- 0.3 lo g_{2} (0.3) - 0.7 lo g_{2} (0.7)

- lo g_{2} (\frac{7 ^{0.7} \cdot 3 ^{0.3}}{5}) + 1

Decimal Notation

0.88129 \dots

Solution steps

- 0.3 lo g_{2} (0.3) - 0.7 lo g_{2} (0.7)

- 0.3 lo g_{2} (0.3) = - 0.3 lo g_{2} (\frac{3}{10})

0.7 = \frac{7}{10}

= - 0.3 lo g_{2} (\frac{3}{10}) - 0.7 lo g_{2} (\frac{7}{10})

- 0.3 lo g_{2} (\frac{3}{10}) = - lo g_{2} ((\frac{3}{10})^{0.3})

- 0.7 lo g_{2} (\frac{7}{10}) = - lo g_{2} ((\frac{7}{10})^{0.7})

= - lo g_{2} ((\frac{3}{10})^{0.3}) - lo g_{2} ((\frac{7}{10})^{0.7})

Apply log rule:

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

- lo g_{2} ((\frac{7}{10})^{0.7}) - lo g_{2} ((\frac{3}{10})^{0.3}) = lo g_{2} ((\frac{7}{10})^{0.7} (\frac{3}{10})^{0.3})

= - lo g_{2} ((\frac{7}{10})^{0.7} (\frac{3}{10})^{0.3})

Simplify

(\frac{7}{10})^{0.7} (\frac{3}{10})^{0.3} : \frac{7 ^{0.7} \cdot 3 ^{0.3}}{10}

= - lo g_{2} (\frac{7 ^{0.7} \cdot 3 ^{0.3}}{10})

Apply log rule:

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

lo g_{2} (\frac{7 ^{0.7} \cdot 3 ^{0.3}}{10}) = lo g_{2} (7^{0.7} \cdot 3^{0.3}) - lo g_{2} (10)

= - (lo g_{2} (7^{0.7} \cdot 3^{0.3}) - lo g_{2} (10))

lo g_{2} (7^{0.7} \cdot 3^{0.3}) = 0.7 lo g_{2} (7) + 0.3 lo g_{2} (3)

= - ((0.7 lo g_{2} (7) + 0.3 lo g_{2} (3)) - lo g_{2} (10))

Apply rule:

(a) = a

(0.7 lo g_{2} (7) + 0.3 lo g_{2} (3)) = 0.7 lo g_{2} (7) + 0.3 lo g_{2} (3)

= - (0.7 lo g_{2} (7) + 0.3 lo g_{2} (3) - lo g_{2} (10))

lo g_{2} (10) = 1 + lo g_{2} (5)

= - (0.7 lo g_{2} (7) + 0.3 lo g_{2} (3) - (1 + lo g_{2} (5)))

Apply the distributive law:

- (a + b) = - a - b

- (1 + lo g_{2} (5)) = - 1 - lo g_{2} (5)

= - (0.7 lo g_{2} (7) + 0.3 lo g_{2} (3) - 1 - lo g_{2} (5))

0.7 lo g_{2} (7) = lo g_{2} (7^{0.7})

0.3 lo g_{2} (3) = lo g_{2} (3^{0.3})

= - (lo g_{2} (7^{0.7}) + lo g_{2} (3^{0.3}) - 1 - lo g_{2} (5))

lo g_{2} (7^{0.7}) + lo g_{2} (3^{0.3}) - lo g_{2} (5) = lo g_{2} (\frac{7 ^{0.7} \cdot 3 ^{0.3}}{5})

= - (lo g_{2} (\frac{7 ^{0.7} \cdot 3 ^{0.3}}{5}) - 1)

Apply the distributive law:

- (a - b) = - a + b

- (lo g_{2} (\frac{7 ^{0.7} \cdot 3 ^{0.3}}{5}) - 1) = - lo g_{2} (\frac{7 ^{0.7} \cdot 3 ^{0.3}}{5}) + 1

= - lo g_{2} (\frac{7 ^{0.7} \cdot 3 ^{0.3}}{5}) + 1

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What is simplify-0.3log_{2}(0.3)-0.7log_{2}(0.7) ?
The solution to simplify-0.3log_{2}(0.3)-0.7log_{2}(0.7) is -log_{2}((7^{0.7}*3^{0.3})/5)+1