What is log_{10}(2)+3log_{10}(x)-1/2 log_{10}(y) ?

The solution to log_{10}(2)+3log_{10}(x)-1/2 log_{10}(y) is log_{10}((2x^3sqrt(y))/y)

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simplify log_{10}(2)+3log_{10}(x)-1/2 log_{10}(y)

simplify

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

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

Solution steps

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

Apply log rule:

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

3 lo g_{10} (x) = lo g_{10} (x^{3})

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

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

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

Simplify

\frac{2 x ^{3}}{y ^{\frac{1}{2}}} : \frac{2 x ^{3} y}{y}

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

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What is log_{10}(2)+3log_{10}(x)-1/2 log_{10}(y) ?
The solution to log_{10}(2)+3log_{10}(x)-1/2 log_{10}(y) is log_{10}((2x^3sqrt(y))/y)