What is simplify-1/3 log_{10}(1/3)-2/3 log_{10}(2/3) ?

The solution to simplify-1/3 log_{10}(1/3)-2/3 log_{10}(2/3) is (log_{10}(27/4))/3

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simplify-1/3 log_{10}(1/3)-2/3 log_{10}(2/3)

simplify

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

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

Decimal Notation

0.27643 \dots

Solution steps

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

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

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

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

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

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

= lo g_{10} \frac{3 ^{\frac{1}{3}}}{( \frac{2}{3} ) ^{\frac{2}{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} (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{lo g _{10} ( 3 ) - lo g _{10} ( \frac{4}{9} )}{3}

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

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

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

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

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

lo g_{2} (225) - lo g_{2} (5) + lo g_{2} (3)

lo g_{3} (x^{2}) + lo g_{3} (25)

3 lo g_{10} (7 x) - 4 lo g_{10} (7 y)

lo g_{2} (\frac{1}{m})

ln (\frac{2}{11})

What is simplify-1/3 log_{10}(1/3)-2/3 log_{10}(2/3) ?
The solution to simplify-1/3 log_{10}(1/3)-2/3 log_{10}(2/3) is (log_{10}(27/4))/3