What is log_{10}(p)-1/3 log_{10}(q) ?

The solution to log_{10}(p)-1/3 log_{10}(q) is log_{10}((pq^{2/3})/q)

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simplify log_{10}(p)-1/3 log_{10}(q)

simplify

lo g_{10} (p) - \frac{1}{3} lo g_{10} (q)

lo g_{10} (\frac{p q ^{\frac{2}{3}}}{q})

Solution steps

lo g_{10} (p) - \frac{1}{3} lo g_{10} (q)

Apply log rule:

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

\frac{1}{3} lo g_{10} (q) = lo g_{10} (q^{\frac{1}{3}})

= lo g_{10} (p) - lo g_{10} (q^{\frac{1}{3}})

Apply log rule:

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

lo g_{10} (p) - lo g_{10} (q^{\frac{1}{3}}) = lo g_{10} (\frac{p}{q ^{\frac{1}{3}}})

= lo g_{10} (\frac{p}{q ^{\frac{1}{3}}})

Simplify

\frac{p}{q ^{\frac{1}{3}}} : \frac{p q ^{\frac{2}{3}}}{q}

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

ln (5) + \frac{1}{2} ln (9)

lo g_{6} (\frac{x ^{3}}{y ^{8}})

ln (5) + \frac{1}{2} ln (4)

What is log_{10}(p)-1/3 log_{10}(q) ?
The solution to log_{10}(p)-1/3 log_{10}(q) is log_{10}((pq^{2/3})/q)