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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

2 ln (x) = ln (x^{2})

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

Apply log rule:

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

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

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

Simplify

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

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

lo g_{e} (\frac{3}{2})

lo g_{2} (\frac{c}{8})

lo g_{\frac{3}{2}} (9) + lo g_{\frac{3}{2}} (3) - lo g_{\frac{3}{2}} (8)

ln (\frac{e ^{2}}{13})

ln (a) + ln (b) - ln (c)

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