What is simplify-3/4 log_{e}(3/4)-1/4 log_{e}(1/4) ?

The solution to simplify-3/4 log_{e}(3/4)-1/4 log_{e}(1/4) is -1/4 ln(27/256)

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simplify-3/4 log_{e}(3/4)-1/4 log_{e}(1/4)

simplify

- \frac{3}{4} lo g_{e} (\frac{3}{4}) - \frac{1}{4} lo g_{e} (\frac{1}{4})

- \frac{1}{4} ln (\frac{27}{256})

Decimal Notation

0.56233 \dots

Solution steps

- \frac{3}{4} lo g_{e} (\frac{3}{4}) - \frac{1}{4} lo g_{e} (\frac{1}{4})

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{3}{4} lo g_{e} (\frac{3}{4}) = lo g_{e} ((\frac{3}{4})^{\frac{3}{4}})

= - lo g_{e} ((\frac{3}{4})^{\frac{3}{4}}) - \frac{1}{4} lo g_{e} (\frac{1}{4})

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{4} lo g_{e} (\frac{1}{4}) = lo g_{e} ((\frac{1}{4})^{\frac{1}{4}})

= - lo g_{e} ((\frac{3}{4})^{\frac{3}{4}}) - lo g_{e} ((\frac{1}{4})^{\frac{1}{4}})

Rewrite

- lo g_{e} ((\frac{3}{4})^{\frac{3}{4}}) - lo g_{e} ((\frac{1}{4})^{\frac{1}{4}})

- (lo g_{e} ((\frac{3}{4})^{\frac{3}{4}}) + lo g_{e} ((\frac{1}{4})^{\frac{1}{4}}))

= - (lo g_{e} ((\frac{3}{4})^{\frac{3}{4}}) + lo g_{e} ((\frac{1}{4})^{\frac{1}{4}}))

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

lo g_{e} ((\frac{3}{4})^{\frac{3}{4}}) + lo g_{e} ((\frac{1}{4})^{\frac{1}{4}}) = lo g_{e} ((\frac{3}{4})^{\frac{3}{4}} (\frac{1}{4})^{\frac{1}{4}})

= - lo g_{e} ((\frac{3}{4})^{\frac{3}{4}} (\frac{1}{4})^{\frac{1}{4}})

Apply log rule:

lo g_{e} (x) = ln (x)

= - ln ((\frac{3}{4})^{\frac{3}{4}} (\frac{1}{4})^{\frac{1}{4}})

Simplify

ln ((\frac{3}{4})^{\frac{3}{4}} (\frac{1}{4})^{\frac{1}{4}}) : \frac{1}{4} ln ((\frac{3}{4})^{3} \frac{1}{4})

= - \frac{1}{4} ln ((\frac{3}{4})^{3} \frac{1}{4})

(\frac{3}{4})^{3} = \frac{3 ^{3}}{4 ^{3}}

= - \frac{1}{4} ln (\frac{1}{4} \cdot \frac{3 ^{3}}{4 ^{3}})

Multiply

\frac{3 ^{3}}{4 ^{3}} \cdot \frac{1}{4} : \frac{27}{256}

= - \frac{1}{4} ln (\frac{27}{256})

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What is simplify-3/4 log_{e}(3/4)-1/4 log_{e}(1/4) ?
The solution to simplify-3/4 log_{e}(3/4)-1/4 log_{e}(1/4) is -1/4 ln(27/256)