What is log_{4}(((1024)/(xy^2))^2) ?

The solution to log_{4}(((1024)/(xy^2))^2) is 10-log_{2}(x)-2log_{2}(y)

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simplify log_{4}(((1024)/(xy^2))^2)

simplify

lo g_{4} ((\frac{1024}{x y ^{2}})^{2})

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

Solution steps

lo g_{4} ((\frac{1024}{x y ^{2}})^{2})

Apply log rule

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

lo g_{4} ((\frac{1024}{x y ^{2}})^{2}) = 2 lo g_{4} (\frac{1024}{x y ^{2}})

= 2 lo g_{4} (\frac{1024}{x y ^{2}})

Factor the number:

4 = 2^{2}

= 2 lo g_{2^{2}} (\frac{1024}{x y ^{2}})

Apply log rule:

lo g_{a^{b}} (x) = \frac{1}{b} lo g_{a} (x), a > 0

lo g_{2^{2}} (\frac{1024}{x y ^{2}}) = \frac{1}{2} lo g_{2} (\frac{1024}{x y ^{2}})

= 2 \cdot \frac{1}{2} lo g_{2} (\frac{1024}{x y ^{2}})

Simplify

2 \cdot \frac{1}{2} lo g_{2} (\frac{1024}{x y ^{2}}) : lo g_{2} (\frac{1024}{x y ^{2}})

= lo g_{2} (\frac{1024}{x y ^{2}})

Apply log rule:

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

lo g_{2} (\frac{1024}{x y ^{2}}) = lo g_{2} (1024) - lo g_{2} (x y^{2})

= lo g_{2} (1024) - lo g_{2} (x y^{2})

lo g_{2} (1024) = 10

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

= 10 - (lo g_{2} (x) + 2 lo g_{2} (y))

Apply the distributive law:

- (a + b) = - a - b

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

= 10 - lo g_{2} (x) - 2 lo g_{2} (y)

x \frac{1}{2}

(46)^{2}

- 2 \cdot - 2

\frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3}

x - 6 + x^{2}

What is log_{4}(((1024)/(xy^2))^2) ?
The solution to log_{4}(((1024)/(xy^2))^2) is 10-log_{2}(x)-2log_{2}(y)