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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Simplify

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

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

\frac{1}{2} lo g_{10} (c) + 7 lo g_{10} (d)

\frac{1}{3} lo g_{k} (x) + 6 lo g_{k} (y) - 4 lo g_{k} (x)

lo g_{4} (8 x)

lo g_{3} (\frac{9}{y})

\frac{1}{2} lo g_{10} (a) + 4 lo g_{10} (b)

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