What is simplify-1/3*log_{2}(1/3)-2/3*log_{2}(2/3) ?

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

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

simplify

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

lo g_{2} \frac{3 3}{3 \frac{4}{9}}

Decimal

0.91829 \dots

Solution steps

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

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

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

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

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

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

Apply log rule:

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

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

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

(\frac{2}{3})^{\frac{2}{3}} = 3 \frac{4}{9}

= lo g_{2} \frac{3 ^{\frac{1}{3}}}{3 \frac{4}{9}}

Apply exponent rule:

a^{\frac{1}{n}} = n a

3^{\frac{1}{3}} = 33

= lo g_{2} \frac{3 3}{3 \frac{4}{9}}

lo g_{10} (y^{2} - 3 y + 2) - lo g_{10} (y^{2} - 1)

ln (\frac{3859}{35109})

lo g_{2} (\frac{64}{250}) + 6 lo g_{4} (5)

lo g_{2} (x y)

lo g_{10} (\frac{x ^{2}}{5})

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