What is simplify-6/9 log_{2}(6/9)-3/9 log_{2}(3/9) ?

The solution to simplify-6/9 log_{2}(6/9)-3/9 log_{2}(3/9) is log_{2}((\sqrt[3]{3})/(\sqrt[3]{4/9)})

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simplify-6/9 log_{2}(6/9)-3/9 log_{2}(3/9)

simplify

- \frac{6}{9} lo g_{2} (\frac{6}{9}) - \frac{3}{9} lo g_{2} (\frac{3}{9})

Decimal Notation

0.91829 \dots

Solution steps

- \frac{6}{9} lo g_{2} (\frac{6}{9}) - \frac{3}{9} lo g_{2} (\frac{3}{9})

- \frac{6}{9} lo g_{2} (\frac{6}{9}) = - \frac{2}{3} lo g_{2} (\frac{2}{3})

\frac{3}{9} lo g_{2} (\frac{3}{9}) = - \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:

4 lo g_{b} (2) + lo g_{b} (5)

lo g_{7} (2) + lo g_{7} (5)

ln (\frac{8}{5 x ^{5} y})

lo g_{3} (\frac{6 y}{z})

ln (\frac{e ^{6}}{7})

What is simplify-6/9 log_{2}(6/9)-3/9 log_{2}(3/9) ?
The solution to simplify-6/9 log_{2}(6/9)-3/9 log_{2}(3/9) is log_{2}((\sqrt[3]{3})/(\sqrt[3]{4/9)})