What is 8/12 log_{10}(8/12)+4/12 log_{10}(4/12) ?

The solution to 8/12 log_{10}(8/12)+4/12 log_{10}(4/12) is (log_{10}(4/27))/3

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simplify 8/12 log_{10}(8/12)+4/12 log_{10}(4/12)

simplify

\frac{8}{12} lo g_{10} (\frac{8}{12}) + \frac{4}{12} lo g_{10} (\frac{4}{12})

\frac{lo g _{10} ( \frac{4}{27} )}{3}

Decimal Notation

- 0.27643 \dots

Solution steps

\frac{8}{12} lo g_{10} (\frac{8}{12}) + \frac{4}{12} lo g_{10} (\frac{4}{12})

\frac{8}{12} lo g_{10} (\frac{8}{12}) = \frac{2}{3} lo g_{10} (\frac{2}{3})

\frac{4}{12} lo g_{10} (\frac{4}{12}) = - \frac{1}{3} lo g_{10} (3)

= \frac{2}{3} lo g_{10} (\frac{2}{3}) - \frac{1}{3} lo g_{10} (3)

\frac{2}{3} lo g_{10} (\frac{2}{3}) = lo g_{10} ((\frac{2}{3})^{\frac{2}{3}})

- \frac{1}{3} lo g_{10} (3) = - lo g_{10} (3^{\frac{1}{3}})

= lo g_{10} ((\frac{2}{3})^{\frac{2}{3}}) - lo g_{10} (3^{\frac{1}{3}})

Apply log rule:

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

lo g_{10} ((\frac{2}{3})^{\frac{2}{3}}) - lo g_{10} (3^{\frac{1}{3}}) = lo g_{10} \frac{( \frac{2}{3} ) ^{\frac{2}{3}}}{3 ^{\frac{1}{3}}}

= lo g_{10} \frac{( \frac{2}{3} ) ^{\frac{2}{3}}}{3 ^{\frac{1}{3}}}

Apply log rule:

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

lo g_{10} (3^{\frac{1}{3}}) = \frac{1}{3} lo g_{10} (3)

= \frac{1}{3} lo g_{10} (\frac{4}{9}) - \frac{1}{3} lo g_{10} (3)

\frac{1}{3} lo g_{10} (\frac{4}{9}) - \frac{1}{3} lo g_{10} (3) = \frac{lo g _{10} ( \frac{4}{9} ) - lo g _{10} ( 3 )}{3}

= \frac{lo g _{10} ( \frac{4}{9} ) - lo g _{10} ( 3 )}{3}

lo g_{10} (\frac{4}{9}) - lo g_{10} (3) = 2 lo g_{10} (2) - 3 lo g_{10} (3)

= \frac{2 lo g _{10} ( 2 ) - 3 lo g _{10} ( 3 )}{3}

2 lo g_{10} (2) - 3 lo g_{10} (3) = lo g_{10} (\frac{4}{27})

= \frac{lo g _{10} ( \frac{4}{27} )}{3}

ln (\frac{x ^{5}}{x ( 1 + x ) ^{3}})

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lo g_{10} (10000 x^{4} y^{6})

What is 8/12 log_{10}(8/12)+4/12 log_{10}(4/12) ?
The solution to 8/12 log_{10}(8/12)+4/12 log_{10}(4/12) is (log_{10}(4/27))/3