What is 1/3 log_{10}(a)-2/5 log_{10}(b) ?

The solution to 1/3 log_{10}(a)-2/5 log_{10}(b) is log_{10}((a^{1/3}b^{3/5})/b)

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simplify 1/3 log_{10}(a)-2/5 log_{10}(b)

simplify

\frac{1}{3} lo g_{10} (a) - \frac{2}{5} lo g_{10} (b)

lo g_{10} (\frac{a ^{\frac{1}{3}} b ^{\frac{3}{5}}}{b})

Solution steps

\frac{1}{3} lo g_{10} (a) - \frac{2}{5} lo g_{10} (b)

Apply log rule:

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

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

= lo g_{10} (a^{\frac{1}{3}}) - \frac{2}{5} lo g_{10} (b)

Apply log rule:

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

\frac{2}{5} lo g_{10} (b) = lo g_{10} (b^{\frac{2}{5}})

= lo g_{10} (a^{\frac{1}{3}}) - lo g_{10} (b^{\frac{2}{5}})

Apply log rule:

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

lo g_{10} (a^{\frac{1}{3}}) - lo g_{10} (b^{\frac{2}{5}}) = lo g_{10} (\frac{a ^{\frac{1}{3}}}{b ^{\frac{2}{5}}})

= lo g_{10} (\frac{a ^{\frac{1}{3}}}{b ^{\frac{2}{5}}})

Simplify

\frac{a ^{\frac{1}{3}}}{b ^{\frac{2}{5}}} : \frac{a ^{\frac{1}{3}} b ^{\frac{3}{5}}}{b}

= lo g_{10} (\frac{a ^{\frac{1}{3}} b ^{\frac{3}{5}}}{b})

lo g_{10} (2) + lo g_{10} (3) - lo g_{10} (5)

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ln (cos (x)) - ln (sin (x))

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What is 1/3 log_{10}(a)-2/5 log_{10}(b) ?
The solution to 1/3 log_{10}(a)-2/5 log_{10}(b) is log_{10}((a^{1/3}b^{3/5})/b)