What is log_{2}(xy^2)-log_{2}(x^2)-2log_{2}(y) ?

The solution to log_{2}(xy^2)-log_{2}(x^2)-2log_{2}(y) is -log_{2}(x)

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simplify log_{2}(xy^2)-log_{2}(x^2)-2log_{2}(y)

simplify

lo g_{2} (x y^{2}) - lo g_{2} (x^{2}) - 2 lo g_{2} (y)

- lo g_{2} (x)

Solution steps

lo g_{2} (x y^{2}) - lo g_{2} (x^{2}) - 2 lo g_{2} (y)

Apply log rule:

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

- 2 lo g_{2} (y) = - lo g_{2} (y^{2})

= lo g_{2} (x y^{2}) - lo g_{2} (x^{2}) - lo g_{2} (y^{2})

Apply log rule:

lo g_{a} (x) - lo g_{a} (y) = lo g_{a} (\frac{x}{y})

lo g_{2} (x y^{2}) - lo g_{2} (x^{2}) = lo g_{2} (\frac{x y ^{2}}{x ^{2}})

= lo g_{2} (\frac{x y ^{2}}{x ^{2}}) - lo g_{2} (y^{2})

Apply log rule:

lo g_{a} (x) - lo g_{a} (y) = lo g_{a} (\frac{x}{y})

lo g_{2} (\frac{x y ^{2}}{x ^{2}}) - lo g_{2} (y^{2}) = lo g_{2} (\frac{\frac{x y ^{2}}{x ^{2}}}{y ^{2}})

= lo g_{2} (\frac{\frac{x y ^{2}}{x ^{2}}}{y ^{2}})

Cancel

\frac{\frac{x y ^{2}}{x ^{2}}}{y ^{2}} : \frac{1}{x}

= lo g_{2} (\frac{1}{x})

Simplify

\frac{1}{x} : x^{- 1}

= lo g_{2} (x^{- 1})

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x), assuming x > 0

lo g_{2} (x^{- 1}) = - 1 \cdot lo g_{2} (x)

= - 1 \cdot lo g_{2} (x)

Apply rule:

1 \cdot a = a

= - lo g_{2} (x)

(\frac{4}{3})^{- 4}

g^{5} = - 243

(- 2 x^{5})^{3}

(x^{2} - x) + (- 4 x + 4)

(3 + i)^{3}

What is log_{2}(xy^2)-log_{2}(x^2)-2log_{2}(y) ?
The solution to log_{2}(xy^2)-log_{2}(x^2)-2log_{2}(y) is -log_{2}(x)