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})

ln (\frac{2}{6})

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

4 lo g_{10} (x) - \frac{1}{2} lo g_{10} (y)

ln (5) + ln (7)

ln (\frac{100}{40})

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)