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

simplify

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

Decimal Notation

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}}}

Apply exponent rule:

lo g_{5} (x^{2} - 1) - lo g_{5} (x - 1)

lo g_{10} (10000 x)

- lo g_{10} (1.8 \times 1 0^{- 5})

lo g_{10} (4 x)

3 lo g_{2} (5) + lo g_{2} (3)

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)})