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}})

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

Rewrite

- 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}}) + 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})

- lo g_{10} (7.4 \cdot 1 0^{- 4})

ln ((x y)^{3})

3 ln (x - 4) + 2 ln (3) - \frac{1}{2} ln (y)

lo g_{b} (\frac{x ^{4} y ^{3}}{z})

ln (\frac{15}{2})

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)