What is log_{10}((\sqrt[3]{xy})/(xy)) ?

The solution to log_{10}((\sqrt[3]{xy})/(xy)) is -2/3 log_{10}(x)-2/3 log_{10}(y)

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simplify log_{10}((\sqrt[3]{xy})/(xy))

simplify

- \frac{2}{3} lo g_{10} (x) - \frac{2}{3} lo g_{10} (y)

Solution steps

Cancel

= lo g_{10} (\frac{1}{x ^{\frac{2}{3}} y ^{\frac{2}{3}}})

Simplify

\frac{1}{x ^{\frac{2}{3}} y ^{\frac{2}{3}}} : (x^{\frac{2}{3}} y^{\frac{2}{3}})^{- 1}

= lo g_{10} ((x^{\frac{2}{3}} y^{\frac{2}{3}})^{- 1})

Apply log rule

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

lo g_{10} ((x^{\frac{2}{3}} y^{\frac{2}{3}})^{- 1}) = - 1 \cdot lo g_{10} (x^{\frac{2}{3}} y^{\frac{2}{3}})

= - 1 \cdot lo g_{10} (x^{\frac{2}{3}} y^{\frac{2}{3}})

Apply rule:

1 \cdot a = a

= - lo g_{10} (x^{\frac{2}{3}} y^{\frac{2}{3}})

Apply log rule:

lo g_{a} (x y) = lo g_{a} (x) + lo g_{a} (y)

- lo g_{10} (x^{\frac{2}{3}} y^{\frac{2}{3}}) = - (lo g_{10} (x^{\frac{2}{3}}) + lo g_{10} (y^{\frac{2}{3}}))

= - (lo g_{10} (x^{\frac{2}{3}}) + lo g_{10} (y^{\frac{2}{3}}))

lo g_{10} (x^{\frac{2}{3}}) = \frac{2}{3} lo g_{10} (x)

lo g_{10} (y^{\frac{2}{3}}) = \frac{2}{3} lo g_{10} (y)

= - (\frac{2}{3} lo g_{10} (x) + \frac{2}{3} lo g_{10} (y))

Apply the distributive law:

- (a + b) = - a - b

- (\frac{2}{3} lo g_{10} (x) + \frac{2}{3} lo g_{10} (y)) = - \frac{2}{3} lo g_{10} (x) - \frac{2}{3} lo g_{10} (y)

= - \frac{2}{3} lo g_{10} (x) - \frac{2}{3} lo g_{10} (y)

ln (x) + ln (x + 8)

lo g_{a} (2 x)

lo g_{10} (\frac{5 x}{y ^{3}})

lo g_{3} (A) + lo g_{3} (B)

15872 ln^{2} (\frac{7}{5})

What is log_{10}((\sqrt[3]{xy})/(xy)) ?
The solution to log_{10}((\sqrt[3]{xy})/(xy)) is -2/3 log_{10}(x)-2/3 log_{10}(y)