What is simplify-0.1log_{2}(0.1)-0.9log_{2}(0.9) ?

The solution to simplify-0.1log_{2}(0.1)-0.9log_{2}(0.9) is -log_{2}(0.72246…)

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simplify-0.1log_{2}(0.1)-0.9log_{2}(0.9)

simplify

- 0.1 lo g_{2} (0.1) - 0.9 lo g_{2} (0.9)

- lo g_{2} (0.72246 \dots)

Decimal Notation

0.46899 \dots

Solution steps

- 0.1 lo g_{2} (0.1) - 0.9 lo g_{2} (0.9)

Apply log rule:

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

0.1 lo g_{2} (0.1) = lo g_{2} (0. 1^{0.1})

= - lo g_{2} (0. 1^{0.1}) - 0.9 lo g_{2} (0.9)

Apply log rule:

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

0.9 lo g_{2} (0.9) = lo g_{2} (0. 9^{0.9})

= - lo g_{2} (0. 1^{0.1}) - lo g_{2} (0. 9^{0.9})

Rewrite

- lo g_{2} (0. 1^{0.1}) - lo g_{2} (0. 9^{0.9})

- (lo g_{2} (0. 1^{0.1}) + lo g_{2} (0. 9^{0.9}))

= - (lo g_{2} (0. 1^{0.1}) + lo g_{2} (0. 9^{0.9}))

Apply log rule:

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

lo g_{2} (0. 1^{0.1}) + lo g_{2} (0. 9^{0.9}) = lo g_{2} (0. 1^{0.1} \cdot 0. 9^{0.9})

= - lo g_{2} (0. 1^{0.1} \cdot 0. 9^{0.9})

0. 1^{0.1} \cdot 0. 9^{0.9} = 0.72246 \dots

= - lo g_{2} (0.72246 \dots)

lo g_{10} (2) + lo g_{10} (x + 3) - lo g_{10} (x + 5)

ln (\frac{10}{x})

lo g_{2} (\frac{x}{5})

ln (\frac{16}{5})

lo g_{10} (5 b) + lo g_{10} (2 c^{2})

What is simplify-0.1log_{2}(0.1)-0.9log_{2}(0.9) ?
The solution to simplify-0.1log_{2}(0.1)-0.9log_{2}(0.9) is -log_{2}(0.72246…)