What is log_{10}(x)-1/4 log_{10}(y)+5log_{10}(z) ?

The solution to log_{10}(x)-1/4 log_{10}(y)+5log_{10}(z) is log_{10}((xz^5y^{3/4})/y)

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simplify log_{10}(x)-1/4 log_{10}(y)+5log_{10}(z)

simplify

lo g_{10} (x) - \frac{1}{4} lo g_{10} (y) + 5 lo g_{10} (z)

lo g_{10} (\frac{x z ^{5} y ^{\frac{3}{4}}}{y})

Solution steps

lo g_{10} (x) - \frac{1}{4} lo g_{10} (y) + 5 lo g_{10} (z)

Apply log rule:

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

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

= lo g_{10} (x) - lo g_{10} (y^{\frac{1}{4}}) + 5 lo g_{10} (z)

Apply log rule:

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

lo g_{10} (x) - lo g_{10} (y^{\frac{1}{4}}) = lo g_{10} (\frac{x}{y ^{\frac{1}{4}}})

= lo g_{10} (\frac{x}{y ^{\frac{1}{4}}}) + 5 lo g_{10} (z)

Simplify

\frac{x}{y ^{\frac{1}{4}}} : \frac{x y ^{\frac{3}{4}}}{y}

= lo g_{10} (\frac{x y ^{\frac{3}{4}}}{y}) + 5 lo g_{10} (z)

Apply log rule:

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

5 lo g_{10} (z) = lo g_{10} (z^{5})

= lo g_{10} (\frac{x y ^{\frac{3}{4}}}{y}) + lo g_{10} (z^{5})

Apply log rule:

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

lo g_{10} (\frac{x y ^{\frac{3}{4}}}{y}) + lo g_{10} (z^{5}) = lo g_{10} (\frac{x y ^{\frac{3}{4}}}{y} z^{5})

= lo g_{10} (\frac{x y ^{\frac{3}{4}}}{y} z^{5})

Simplify

\frac{x y ^{\frac{3}{4}}}{y} z^{5} : \frac{x z ^{5} y ^{\frac{3}{4}}}{y}

= lo g_{10} (\frac{x z ^{5} y ^{\frac{3}{4}}}{y})

ln (\frac{1}{x + 3})

ln (x^{3} e^{- x^{2}})

ln (\frac{12 x ^{5}}{7})

What is log_{10}(x)-1/4 log_{10}(y)+5log_{10}(z) ?
The solution to log_{10}(x)-1/4 log_{10}(y)+5log_{10}(z) is log_{10}((xz^5y^{3/4})/y)