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

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

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

simplify

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

- lo g_{2} (0.72246 \dots)

Decimal Notation

0.46899 \dots

Solution steps

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

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

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

Rewrite

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

Apply log rule:

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

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

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

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

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

lo g_{m} (\frac{a \cdot b \cdot c ^{\frac{1}{4}}}{d \cdot e ^{\frac{1}{2}}})

ln (4 x^{3})

ln (2 x (x + 3)^{5})

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

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