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

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

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

simplify

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

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}})

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

Rewrite as

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

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

assuming

x \geq 0

= \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})

ln \frac{( \frac{7 + 3 5}{2} )}{6}

ln (\frac{1}{96})

ln (x - 10) - ln (x - 1)

- ln (\frac{1}{a})

lo g_{10} ((8^{34}) (2^{43}) (1 6^{68}))

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