What is 8/12 log_{2}(8/12)+4/12 log_{2}(4/12) ?

The solution to 8/12 log_{2}(8/12)+4/12 log_{2}(4/12) is log_{2}((\sqrt[3]{4/9})/(\sqrt[3]{3)})

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simplify 8/12 log_{2}(8/12)+4/12 log_{2}(4/12)

simplify

\frac{8}{12} lo g_{2} (\frac{8}{12}) + \frac{4}{12} lo g_{2} (\frac{4}{12})

Decimal Notation

- 0.91829 \dots

Solution steps

\frac{8}{12} lo g_{2} (\frac{8}{12}) + \frac{4}{12} lo g_{2} (\frac{4}{12})

\frac{8}{12} lo g_{2} (\frac{8}{12}) = \frac{2}{3} lo g_{2} (\frac{2}{3})

\frac{4}{12} lo g_{2} (\frac{4}{12}) = - \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:

lo g_{10} ((19 y)^{6})

lo g_{3} (27 (x^{2} - 49))

lo g_{2} (9) - lo g_{2} (y) + \frac{1}{2} lo g_{2} (x)

\frac{1}{18} ln (\frac{167}{23})

lo g_{10} (A) + lo g_{10} (B) - lo g_{10} (\frac{1}{C})

What is 8/12 log_{2}(8/12)+4/12 log_{2}(4/12) ?
The solution to 8/12 log_{2}(8/12)+4/12 log_{2}(4/12) is log_{2}((\sqrt[3]{4/9})/(\sqrt[3]{3)})