What is 2/3 ln(x)-1/2 ln(y) ?

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

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simplify 2/3 ln(x)-1/2 ln(y)

simplify

\frac{2}{3} ln (x) - \frac{1}{2} ln (y)

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

Solution steps

\frac{2}{3} ln (x) - \frac{1}{2} ln (y)

Apply log rule:

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

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

= ln (x^{\frac{2}{3}}) - \frac{1}{2} ln (y)

Apply log rule:

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

\frac{1}{2} ln (y) = ln (y^{\frac{1}{2}})

= ln (x^{\frac{2}{3}}) - ln (y^{\frac{1}{2}})

Apply log rule:

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

ln (x^{\frac{2}{3}}) - ln (y^{\frac{1}{2}}) = ln (\frac{x ^{\frac{2}{3}}}{y ^{\frac{1}{2}}})

= ln (\frac{x ^{\frac{2}{3}}}{y ^{\frac{1}{2}}})

Simplify

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

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

ln (\frac{1}{x ^{n}})

lo g_{3} (6) - lo g_{3} (4)

lo g_{10} (\frac{6}{x})

lo g_{3} (7 x)

ln (x + 3) + 2 ln (x)

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