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

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

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

simplify

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

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

Decimal Notation

0.56233 \dots

Solution steps

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

Rewrite

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

Apply log rule:

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

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

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

Apply log rule:

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

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

Simplify

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

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

- ln (\frac{10 \cdot e ^{- x} + 40 - 4}{10})

ln (\frac{t}{x})

- lo g_{10} (3.16 \cdot 1 0^{- 9})

ln (\frac{l}{x !})

lo g_{10} (5^{3 - x})

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