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

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

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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))

Apply rule:

a - (- b) = a + b

- \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}) + \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} (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:

2 (lo g_{5} (x) + 2 lo g_{5} (y) - 3 lo g_{5} (z))

lo g_{b} (2 x)

lo g_{2} (A) + lo g_{2} (B) - 2 lo g_{2} (C)

ln (\frac{3}{5})

lo g_{10} (x) + lo g_{10} (y)

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