What is log_{2}(0.7)*0.7+log_{2}(0.3)*0.3 ?

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

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

simplify

lo g_{2} (0.7) \cdot 0.7 + lo g_{2} (0.3) \cdot 0.3

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

Decimal Notation

- 0.88129 \dots

Solution steps

lo g_{2} (0.7) \cdot 0.7 + lo g_{2} (0.3) \cdot 0.3

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

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

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

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

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

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

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

- ln (\frac{( x - y )}{x})

6 lo g_{b} (3 w + 1) + \frac{1}{4} lo g_{b} (w + 9)

ln (\frac{θ ^{3}}{a ( 3 )} 2^{2} e^{2 θ})

lo g_{e} (2 π)

- 2 ln (\frac{3}{4}) + ln (\frac{5}{9})

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