What is 1/3 log_{b}(x^4y^5)-3/4 log_{b}(x^2y) ?

The solution to 1/3 log_{b}(x^4y^5)-3/4 log_{b}(x^2y) is log_{b}((\sqrt[12]{y^{11}})/(\sqrt[6]{x)})

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simplify 1/3 log_{b}(x^4y^5)-3/4 log_{b}(x^2y)

simplify

\frac{1}{3} lo g_{b} (x^{4} y^{5}) - \frac{3}{4} lo g_{b} (x^{2} y)

Solution steps

\frac{1}{3} lo g_{b} (x^{4} y^{5}) - \frac{3}{4} lo g_{b} (x^{2} y)

\frac{1}{3} lo g_{b} (x^{4} y^{5}) = lo g_{b} ((x^{4} y^{5})^{\frac{1}{3}})

- \frac{3}{4} lo g_{b} (x^{2} y) = - lo g_{b} ((x^{2} y)^{\frac{3}{4}})

= lo g_{b} ((x^{4} y^{5})^{\frac{1}{3}}) - lo g_{b} ((x^{2} y)^{\frac{3}{4}})

Apply log rule:

lo g_{a} (x) - lo g_{a} (y) = lo g_{a} (\frac{x}{y})

lo g_{b} ((x^{4} y^{5})^{\frac{1}{3}}) - lo g_{b} ((x^{2} y)^{\frac{3}{4}}) = lo g_{b} \frac{( x ^{4} y ^{5} ) ^{\frac{1}{3}}}{( x ^{2} y ) ^{\frac{3}{4}}}

= lo g_{b} \frac{( x ^{4} y ^{5} ) ^{\frac{1}{3}}}{( x ^{2} y ) ^{\frac{3}{4}}}

Cancel

\frac{( x ^{4} y ^{5} ) ^{\frac{1}{3}}}{( x ^{2} y ) ^{\frac{3}{4}}} : \frac{y ^{\frac{11}{12}}}{x ^{\frac{1}{6}}}

= lo g_{b} (\frac{y ^{\frac{11}{12}}}{x ^{\frac{1}{6}}})

Apply exponent rule:

ln (\frac{89}{2})

lo g_{5} (\frac{125 b ^{5} y ^{2} - 8}{y ( x + 8 ) ^{2}})

ln (2 x) - 3

lo g_{7} (\frac{x ^{4}}{y ^{2}})

ln (1 + e^{x} - 1) - ln (1 + e^{x} + 1)

What is 1/3 log_{b}(x^4y^5)-3/4 log_{b}(x^2y) ?
The solution to 1/3 log_{b}(x^4y^5)-3/4 log_{b}(x^2y) is log_{b}((\sqrt[12]{y^{11}})/(\sqrt[6]{x)})