What is 9ln(x)-1/3 ln(y) ?

The solution to 9ln(x)-1/3 ln(y) is ln((x^9y^{2/3})/y)

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simplify 9ln(x)-1/3 ln(y)

simplify

9 ln (x) - \frac{1}{3} ln (y)

ln (\frac{x ^{9} y ^{\frac{2}{3}}}{y})

Solution steps

9 ln (x) - \frac{1}{3} ln (y)

Apply log rule:

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

9 ln (x) = ln (x^{9})

= ln (x^{9}) - \frac{1}{3} ln (y)

Apply log rule:

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

\frac{1}{3} ln (y) = ln (y^{\frac{1}{3}})

= ln (x^{9}) - ln (y^{\frac{1}{3}})

Apply log rule:

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

ln (x^{9}) - ln (y^{\frac{1}{3}}) = ln (\frac{x ^{9}}{y ^{\frac{1}{3}}})

= ln (\frac{x ^{9}}{y ^{\frac{1}{3}}})

Simplify

\frac{x ^{9}}{y ^{\frac{1}{3}}} : \frac{x ^{9} y ^{\frac{2}{3}}}{y}

= ln (\frac{x ^{9} y ^{\frac{2}{3}}}{y})

lo g_{b} (\frac{p ^{4} q ^{6}}{m ^{4} b ^{9}})

lo g_{10} (\frac{10}{x ^{2} y})

(\frac{1}{2})^{3} - 24 ln (\frac{1}{2}) + 8

(\frac{1}{2})^{3} - 24 ln (\frac{1}{2}) + 3

ln (\frac{5000}{200})

What is 9ln(x)-1/3 ln(y) ?
The solution to 9ln(x)-1/3 ln(y) is ln((x^9y^{2/3})/y)