What is 5/14 log_{2}(5/14)+9/14 log_{2}(9/14) ?

The solution to 5/14 log_{2}(5/14)+9/14 log_{2}(9/14) is 1/14 log_{2}((1210689028125)/(14^{14)})

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simplify 5/14 log_{2}(5/14)+9/14 log_{2}(9/14)

simplify

\frac{5}{14} lo g_{2} (\frac{5}{14}) + \frac{9}{14} lo g_{2} (\frac{9}{14})

\frac{1}{14} lo g_{2} (\frac{1210689028125}{1 4 ^{14}})

Decimal Notation

- 0.94028 \dots

Solution steps

\frac{5}{14} lo g_{2} (\frac{5}{14}) + \frac{9}{14} lo g_{2} (\frac{9}{14})

Apply log rule:

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

\frac{5}{14} lo g_{2} (\frac{5}{14}) = lo g_{2} ((\frac{5}{14})^{\frac{5}{14}})

= lo g_{2} ((\frac{5}{14})^{\frac{5}{14}}) + \frac{9}{14} lo g_{2} (\frac{9}{14})

Apply log rule:

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

\frac{9}{14} lo g_{2} (\frac{9}{14}) = lo g_{2} ((\frac{9}{14})^{\frac{9}{14}})

= lo g_{2} ((\frac{5}{14})^{\frac{5}{14}}) + lo g_{2} ((\frac{9}{14})^{\frac{9}{14}})

Apply log rule:

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

lo g_{2} ((\frac{5}{14})^{\frac{5}{14}}) + lo g_{2} ((\frac{9}{14})^{\frac{9}{14}}) = lo g_{2} ((\frac{5}{14})^{\frac{5}{14}} (\frac{9}{14})^{\frac{9}{14}})

= lo g_{2} ((\frac{5}{14})^{\frac{5}{14}} (\frac{9}{14})^{\frac{9}{14}})

Apply exponent rule:

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

Rewrite as

= lo g_{2} ((\frac{5}{14})^{5} (\frac{9}{14})^{9})^{\frac{1}{14}}

Apply log rule

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

assuming

x \geq 0

= \frac{1}{14} lo g_{2} ((\frac{9}{14})^{9} (\frac{5}{14})^{5})

(\frac{5}{14})^{5} = \frac{5 ^{5}}{1 4 ^{5}}

= \frac{1}{14} lo g_{2} ((\frac{9}{14})^{9} \frac{5 ^{5}}{1 4 ^{5}})

(\frac{9}{14})^{9} = \frac{9 ^{9}}{1 4 ^{9}}

= \frac{1}{14} lo g_{2} (\frac{5 ^{5}}{1 4 ^{5}} \cdot \frac{9 ^{9}}{1 4 ^{9}})

Multiply

\frac{5 ^{5}}{1 4 ^{5}} \cdot \frac{9 ^{9}}{1 4 ^{9}} : \frac{1210689028125}{1 4 ^{14}}

= \frac{1}{14} lo g_{2} (\frac{1210689028125}{1 4 ^{14}})

- 3 ln (\frac{4}{9})

4 ln (x + 6) - 2 ln (x)

lo g_{10} (0.1) - lo g_{10} (0.2)

- lo g_{10} (4.7 \cdot 1 0^{- 6})

- lo g_{10} (7.45 \cdot 1 0^{- 6})

What is 5/14 log_{2}(5/14)+9/14 log_{2}(9/14) ?
The solution to 5/14 log_{2}(5/14)+9/14 log_{2}(9/14) is 1/14 log_{2}((1210689028125)/(14^{14)})