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

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

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

simplify

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

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

Decimal Notation

0.65002 \dots

Solution steps

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

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

Rewrite

- 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}}) + 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}}))

Apply log rule:

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

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}} (\frac{1}{6})^{\frac{1}{6}})

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

Simplify

lo g_{2} ((\frac{5}{6})^{\frac{5}{6}} (\frac{1}{6})^{\frac{1}{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{5 ^{5}}{6 ^{5}} \cdot \frac{1}{6} : \frac{3125}{46656}

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

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