What is simplify-(1/6 log_{2}(1/6)+5/6 log_{2}(5/6)) ?

The solution to simplify-(1/6 log_{2}(1/6)+5/6 log_{2}(5/6)) is -1/6 log_{2}(3125/46656)

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simplify-(1/6 log_{2}(1/6)+5/6 log_{2}(5/6))

simplify

- (\frac{1}{6} lo g_{2} (\frac{1}{6}) + \frac{5}{6} lo g_{2} (\frac{5}{6}))

- \frac{1}{6} lo g_{2} (\frac{3125}{46656})

Decimal Notation

0.65002 \dots

Solution steps

- (\frac{1}{6} lo g_{2} (\frac{1}{6}) + \frac{5}{6} lo g_{2} (\frac{5}{6}))

Apply log rule:

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

\frac{1}{6} lo g_{2} (\frac{1}{6}) = lo g_{2} ((\frac{1}{6})^{\frac{1}{6}})

= - (\frac{5}{6} lo g_{2} (\frac{5}{6}) + lo g_{2} ((\frac{1}{6})^{\frac{1}{6}}))

Apply log rule:

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

\frac{5}{6} lo g_{2} (\frac{5}{6}) = lo g_{2} ((\frac{5}{6})^{\frac{5}{6}})

= - (lo g_{2} ((\frac{1}{6})^{\frac{1}{6}}) + lo g_{2} ((\frac{5}{6})^{\frac{5}{6}}))

Apply log rule:

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

lo g_{2} ((\frac{1}{6})^{\frac{1}{6}}) + lo g_{2} ((\frac{5}{6})^{\frac{5}{6}}) = lo g_{2} ((\frac{1}{6})^{\frac{1}{6}} (\frac{5}{6})^{\frac{5}{6}})

= - lo g_{2} ((\frac{1}{6})^{\frac{1}{6}} (\frac{5}{6})^{\frac{5}{6}})

Simplify

lo g_{2} ((\frac{1}{6})^{\frac{1}{6}} (\frac{5}{6})^{\frac{5}{6}}) : \frac{1}{6} lo g_{2} ((\frac{5}{6})^{5} \frac{1}{6})

= - \frac{1}{6} lo g_{2} ((\frac{5}{6})^{5} \frac{1}{6})

(\frac{5}{6})^{5} = \frac{5 ^{5}}{6 ^{5}}

= - \frac{1}{6} lo g_{2} (\frac{1}{6} \cdot \frac{5 ^{5}}{6 ^{5}})

Multiply

\frac{1}{6} \cdot \frac{5 ^{5}}{6 ^{5}} : \frac{3125}{46656}

= - \frac{1}{6} lo g_{2} (\frac{3125}{46656})

y [(2 x^{2} + x) (5 x)]

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- sin ((k + \frac{1}{2}) π)

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- 2 (e^{x} + e^{x} x)

What is simplify-(1/6 log_{2}(1/6)+5/6 log_{2}(5/6)) ?
The solution to simplify-(1/6 log_{2}(1/6)+5/6 log_{2}(5/6)) is -1/6 log_{2}(3125/46656)