What is 2/3 log_{2}(2/3)+1/3 log_{2}(1/3) ?

The solution to 2/3 log_{2}(2/3)+1/3 log_{2}(1/3) is log_{2}((\sqrt[3]{4/9})/(\sqrt[3]{3)})

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simplify 2/3 log_{2}(2/3)+1/3 log_{2}(1/3)

simplify

\frac{2}{3} lo g_{2} (\frac{2}{3}) + \frac{1}{3} lo g_{2} (\frac{1}{3})

Decimal Notation

- 0.91829 \dots

Solution steps

\frac{2}{3} lo g_{2} (\frac{2}{3}) + \frac{1}{3} lo g_{2} (\frac{1}{3})

\frac{1}{3} lo g_{2} (\frac{1}{3}) = - \frac{1}{3} lo g_{2} (3)

= \frac{2}{3} lo g_{2} (\frac{2}{3}) - \frac{1}{3} lo g_{2} (3)

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

- \frac{1}{3} lo g_{2} (3) = - lo g_{2} (3^{\frac{1}{3}})

= lo g_{2} ((\frac{2}{3})^{\frac{2}{3}}) - lo g_{2} (3^{\frac{1}{3}})

Apply log rule:

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

lo g_{2} ((\frac{2}{3})^{\frac{2}{3}}) - lo g_{2} (3^{\frac{1}{3}}) = lo g_{2} \frac{( \frac{2}{3} ) ^{\frac{2}{3}}}{3 ^{\frac{1}{3}}}

= lo g_{2} \frac{( \frac{2}{3} ) ^{\frac{2}{3}}}{3 ^{\frac{1}{3}}}

Apply exponent rule:

(x + 7)^{2} = - 6

3 x^{2} - 23 x - 8 = 0

- 5 t + 9 = 24

2 - 3 (z - 5) + 11 = 4

What is 2/3 log_{2}(2/3)+1/3 log_{2}(1/3) ?
The solution to 2/3 log_{2}(2/3)+1/3 log_{2}(1/3) is log_{2}((\sqrt[3]{4/9})/(\sqrt[3]{3)})