What is 1/4 log_{10}(a)-1/5 log_{10}(b) ?

The solution to 1/4 log_{10}(a)-1/5 log_{10}(b) is log_{10}((a^{1/4}b^{4/5})/b)

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simplify 1/4 log_{10}(a)-1/5 log_{10}(b)

simplify

\frac{1}{4} lo g_{10} (a) - \frac{1}{5} lo g_{10} (b)

lo g_{10} (\frac{a ^{\frac{1}{4}} b ^{\frac{4}{5}}}{b})

Solution steps

\frac{1}{4} lo g_{10} (a) - \frac{1}{5} lo g_{10} (b)

Apply log rule:

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

\frac{1}{4} lo g_{10} (a) = lo g_{10} (a^{\frac{1}{4}})

= lo g_{10} (a^{\frac{1}{4}}) - \frac{1}{5} lo g_{10} (b)

Apply log rule:

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

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

= lo g_{10} (a^{\frac{1}{4}}) - lo g_{10} (b^{\frac{1}{5}})

Apply log rule:

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

lo g_{10} (a^{\frac{1}{4}}) - lo g_{10} (b^{\frac{1}{5}}) = lo g_{10} (\frac{a ^{\frac{1}{4}}}{b ^{\frac{1}{5}}})

= lo g_{10} (\frac{a ^{\frac{1}{4}}}{b ^{\frac{1}{5}}})

Simplify

\frac{a ^{\frac{1}{4}}}{b ^{\frac{1}{5}}} : \frac{a ^{\frac{1}{4}} b ^{\frac{4}{5}}}{b}

= lo g_{10} (\frac{a ^{\frac{1}{4}} b ^{\frac{4}{5}}}{b})

ln (\frac{6}{x})

ln (\frac{x y}{z})

lo g_{b} (y^{3} x)

- lo g_{10} (7.9 \cdot 1 0^{- 5})

\frac{1}{2} lo g_{c} (x) + 4 lo g_{c} (y) - 2 lo g_{c} (x)

What is 1/4 log_{10}(a)-1/5 log_{10}(b) ?
The solution to 1/4 log_{10}(a)-1/5 log_{10}(b) is log_{10}((a^{1/4}b^{4/5})/b)