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

The solution to 3ln(x)-1/9 ln(y) is ln((x^3y^{8/9})/y)

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

simplify

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

ln (\frac{x ^{3} y ^{\frac{8}{9}}}{y})

Solution steps

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

Apply log rule:

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

3 ln (x) = ln (x^{3})

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

\frac{x ^{3}}{y ^{\frac{1}{9}}} : \frac{x ^{3} y ^{\frac{8}{9}}}{y}

= ln (\frac{x ^{3} y ^{\frac{8}{9}}}{y})

- \frac{4}{7} lo g_{2} (\frac{4}{7}) - \frac{3}{7} lo g_{2} (\frac{3}{7})

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

lo g_{4} (6) + 2 lo g_{4} (5)

lo g_{10} (10 x)

ln (x) + ln (9)

What is 3ln(x)-1/9 ln(y) ?
The solution to 3ln(x)-1/9 ln(y) is ln((x^3y^{8/9})/y)